Zonal methods for computation of particle interactions

ABSTRACT

A generalized approach to particle interaction can confer advantages over previously described method in terms of one or more of communications bandwidth and latency and memory access characteristics. These generalizations can involve one or more of at least spatial decomposition, import region rounding, and multiple zone communication scheduling. An architecture for computation of particle interactions makes use various forms of parallelism. In one implementation, the parallelism involves using multiple computation nodes arranged according to a geometric partitioning of a simulation volume.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of

-   -   International Application Serial No. PCT/US2006/032498, filed on        Aug. 18, 2006, designating the United States, and published on        Feb. 22, 2007, as WO 2007/022469 A2, which claims the benefit of    -   International Application Serial No. PCT/US2006/014782, filed        Apr. 19, 2006, designating the United States, and published on        Oct. 26, 2006, as WO 2006/113825 A2, which claims the benefit of    -   U.S. Provisional Application Ser. No. 60/672,717, filed Apr. 19,        2005,    -   U.S. Provisional Application Ser. No. 60/709,184, filed Aug. 18,        2005, and of    -   U.S. Provisional Application Ser. No. 60/756,448, filed Jan. 4,        2006, all of which are incorporated herein by reference.

This application is also related to

-   -   U.S. Provisional Application Ser. No. 60/584,032, filed Jun. 30,        2004,    -   Ser. No. 60/584,000, filed Jun. 30, 2004,    -   U.S. application Ser. No. 11/171,619, filed Jun. 30, 2005,    -   U.S. application Ser. No. 11/171,634, filed Jun. 30, 2005, and        to    -   International Application Serial No. PCT/US2005/023184, filed        Jun. 30, 2005, designating the United States, and published on        Jan. 12, 2006, as WO 2006/004877 A2,        all of which are incorporated herein by reference.

BACKGROUND

This document relates to computation of particle interactions, includingalgorithms and hardware and software techniques that are applicable tosuch computation.

Simulation of multiple-body interactions (often called “N-body”problems) is useful in a number of problem areas including celestialdynamics and computational chemistry. Biomolecular or electrostaticparticle interaction simulations complement experiments by providing auniquely detailed picture of the interaction between particles in asystem. An important issue in simulations is the simulation speed.

Determining the interaction between all pairs of bodies in the system byenumerating the pairs can be computationally intensive, and thus,alternate interaction schemes are often used. For example, all particleswithin a predefined radius of a particular particle are interacted withthat particle while particles farther away are ignored. Special-purposehardware has also been applied in order to reduce the overallcomputation time required for simulation.

SUMMARY

In one aspect, in general, a method for performing computationsassociated with bodies located in a computation region involvesperforming the computations in each subset of multiple subsets of thecomputations. Performing the computations in a subset includes acceptingdata of bodies located in each of a multiple import regions that areassociated with the subset of the computations. The import regions areparts of the computation region. For each combination of a predeterminedset of combinations of multiple of the import regions, computations thatare associated with sets of bodies are performed. For each of these setsof bodies, at least one body of the set is located in the each importregion of the combination.

Aspects can include one or more of the following features.

At least some of the combinations of multiple of the import regions eachconsists of a pair of the import regions.

Each of the subsets of computations is performed on a different node ofa multiple node computing system.

Accepting the data for bodies from the import regions includes acceptingdata from a communication medium coupling at least two of the nodes ofthe computing system.

Each of the subsets of computations is performed in sequence on acomputing system.

The subsets of computations are selected according to memory limitationsof the computing system.

The method includes associating each of the subsets of computations withan iteration in an iterative sequence of computation phases.

Accepting the data for bodies from the import regions includes loadingthe data into a local memory associated with a processor on which thecomputations for the iteration are performed.

Each of the subsets of computations is associated with a separatesub-region of the computation region.

For each of the subsets of computations, the import regions are separatefrom the sub-region of the computation region with which the subset isassociated.

The sub-regions associated with the multiple subsets of computationsform a partition of the computation region. For example, the sub-regionsform a regular partition of the computation region and/or a rectilinearpartition of the computation region.

Performing the computations in each subset of computations furtherincludes performing computations associated with sets of bodies. Foreach of these sets, each body of the set is located in the sub-regionassociated with the subset of computations. The method also includesperforming computations associated with sets of bodies. For each ofthese sets, one body is located in the sub-region associated with thesubset of computations and the other body or bodies of the set arelocated in the import regions associated with the subset ofcomputations.

The performing of computations associated with sets of bodies in whicheach body of the set of bodies is located in the sub-region associatedwith the subset of computations is carried out while accepting the datafor bodies from the import regions.

Performing the computations in each subset of computations furtherincludes performing computations associated with sets of bodies. Foreach of these sets, each body of the set is located in one of the importregions associated with the subset of computations, and no body of theset is located in the sub-region associated with the subset ofcomputations.

Performing the computations in each subset of computations includesapplying a schedule of computations specified according to the importregions.

Each of the computations in the set of computations includes acomputation of data characterizing an interaction of the set of bodiesassociated with that computation.

Each of the computations in the set of computations includes recordingin a data structure an identification of the set of bodies associatedwith that computation.

The method further includes performing computations of datacharacterizing an interaction of each of the sets of bodies recorded inthe data structure.

In performing the computations in all the subsets of computations, forany set of bodies, a computation for that set of bodies is performed inat most one of the subsets.

For at least some of the selected set of combinations of multiple importregions, a computation is performed for every combination of one body ineach of the multiple import regions.

In performing the computations in all the subsets of computations, forany set of bodies, a computation for that set of bodies is performed inat most one of the subsets.

The steps are repeated for each of a series of time steps, thecomputations performed at each time step being used to update locationsof the bodies in the computation region for use in a subsequent timestep.

In another aspect, in general, a method is for performing a set ofcomputations associated with bodies located in a computation region.Each of the computations in the set of computations is associated with apair of the bodies. The method includes accepting data for bodieslocated in a neighborhood. The neighborhood includes a set of multiplezones. Computations associated with pair of bodies are performed suchthat the bodies of each pair are located in different of the zones. Aspatial extent of at least one of the zones is determined to eliminateat least some points in one of the zones that are further away than aminimum distance from all points in another of the zones.

Aspects can include one or more of the following features.

The spatial extent is determined according to a threshold distancebetween pairs of bodies for which computations are to be performed.

A spatial boundary of one of the plurality of zones is determinedaccording to a minimum distance to points within another of the set ofzones.

At least one of the set of zones includes a non-planar boundary.

The computation region is partitioned into regular regions, and at leastone of the set of zones includes some but not all of one of the regularregions.

The computation region is partitioned into regular regions and eachregion is associated with a subset of the computations.

The neighborhood corresponds to one of the regular regions.

In another aspect, in general, a method is for performing a set ofcomputations associated with bodies located in a computation region. Thecomputation region is partitioned into regular regions and each of thecomputations in the set of computations is associated with a set of thebodies. The method includes, for at least some of the regular regions,accepting data for bodies located in a neighborhood of the regularregion, the neighborhood including a set of multiple zones. The set ofzones includes at least one zone that is not contiguous with its partsbeing distributed in at least two dimensions of the computation region.

Each computation in the set of computations is associated with a pair ofbodies.

The neighborhood consists of two zones, and for each of the pairs ofbodies, one body of the pair is located in a different one of the twozones.

The set of zones includes at least one contiguous zone.

The contiguous zone is contiguous with the regular region.

At least two of the zones are elongated in one dimension of thecomputation region.

At least one of the zones extends over at least one other dimension ofthe computation region.

In another aspect, in general, a method is for performing computationsassociated with bodies located in a computation region. The method usesa processor coupled to a local memory. The method includes iterativelyperforming multiple subsets of the computations. In each iteration onesubset of the multiple subsets of computations is performed. Performingthe subset of computations includes loading into the local memory dataof bodies located in each of a set of import regions associated with thesubset of computations. The import regions are parts of the computationregion. Computations associated with pairs of bodies are performed onthe processor. For each pair, one body of the pair is located in a firstof the import region and the other body of the pair is located in asecond of the import region.

Aspects can include one or more of the following features.

The performing the computations associated with pairs of bodies isperformed for each of a predetermined set of combinations of a first anda second of the import regions.

Performing the computations associate with the pairs of bodies includesperforming computations associated with substantially all pairs ofbodies in which one body of the pair is located in the first of theimport regions and the other body of the pair is located in the secondof the import regions.

In another aspect, in general, a method for performing computationsassociated with bodies located in a computation region includes, foreach of a plurality of sub-regions of the computation region,maintaining data for bodies in that sub-region in a correspondingseparate storage area. The sub-regions overlap such that at least somebodies have data maintained in multiple of the storage areas. For eachof the sub-regions, accessing the data for bodies in that sub-regionfrom the corresponding storage area and performing computationsassociated with sets of two or more of the bodies in that sub-region.

Each of the separate storage areas is associated with a node of amultiple node computing system.

Performing the computations associated with pairs of the bodies in thesub-region is performed on the node associated with the storage areacorresponding to the sub-region.

Each of the separate storage areas is associated with a different node.

Each of the sub-regions includes a home region and a region extending atleast a predetermined distance from the home region.

The home regions form a partition of the computation region.

The predetermined distance is at least half an interaction radius forpairwise interactions between bodies.

Performing the computations associated with the pairs of bodies in thesub-region includes performing computations for pairs of bodies forwhich a midpoint of locations of the bodies is within the home regionfor the sub-region.

In another aspect, in general, a method for computing interactions amongsets of bodies located in a computation region includes, for eachcomputation associated with one of the sets of bodies determining acomputation unit of a set of computation units for performing thecomputation according to a mid location of the set of bodies determinedfrom a location of each of the bodies.

Aspects can include one or more of the following features.

The mid location is determined according to a geometric midpoint of thelocations of each of the bodies.

The mid location is determined according to a center of a minimal spherebounding the locations of each of the bodies.

Each of the computation units is associated with a node of a multiplenode computing system.

Each of the computation units is associated with a different node.

Each of the computation units is associated with one of the nodes.

The computation region is partitioned into regular regions eachassociated with a different one of the computation units.

Determining the computation unit for performing a computation associatedwith at least some of the sets of bodies is according to whether the midlocation of the set is within the regular region associated with thecomputation unit.

Determining the computation unit for performing a computation associatedwith at least some of the sets of bodies includes selecting acomputation unit from a set of computation units that include thecomputation unit for which the mid location of the set falls within theregular region associated with the computation unit.

The set of computation units further includes at least one othercomputation unit such that each of the bodies in the set belongs to atleast one other set of bodies with a mid location falling in the regularregion of the at least one other computation unit.

Selecting the computation unit is performed in part according todistribution of computation load across the computation units.

In another aspect, in general, a method is for performing a set ofcomputations associated with bodies located in a computation region.Each of the computations in the set of computations is associated with aset of bodies. The method includes, for at least some of the subsets ofcomputations, performing the computations in that subset ofcomputations. This performing of the computations in that subsetincludes accepting data of bodies located in a vicinity of thesub-region associated with the subset of computations and performingcomputations for the sets of bodies. Each body of the set is located inthe vicinity of the sub-region or in the sub-region. The locations ofthe bodies of the set and the sub-region satisfy a specified criterion.The specified criterion for the locations of the bodies in a set and thesub-region requires that a mid locations of the bodies in the set fallwithin the sub-region.

Aspects can include one or more of the following features.

The mid locations of the bodies in the set is determined according tothe center of a bounding sphere enclosing the locations.

The mid locations of the bodies in the set is determined according to anaverage of the locations.

At least some of the sets of bodies consists of two bodies.

At least some of the sets of bodies consists of three bodies.

At least some of the sets of bodies include more than three bodies.

In another aspect, in general, a method is for performing a set ofcomputations associated with bodies located in a computation region.Each of the computations in the set of computations is associated with apair of bodies. The method includes, for at least some of the subsets ofcomputations, performing the computations in that subset ofcomputations. The performing of the computations in that subset includesaccepting data of bodies located in a vicinity of a sub-regionassociated with the subset of computations and performing computationsfor pairs of bodies. Each body of the pair is located in the vicinity ofthe sub-region or in the sub-region. The locations of the bodies of thepair and the sub-region satisfy a specified criterion.

Aspects can include one or more of the following features.

The specified criterion incorporates information about a distribution ofbodies of pairs in a vicinity of neighboring sub-regions.

The specified criterion for the locations of the bodies in a pair andthe sub-region requires that a midpoint of the locations of the bodiesin the pair fall within the sub-region.

In another aspect, in general, a method for computation of moleculardynamics simulation of a plurality of processing nodes arranged ingeometric grid includes exchanging particle data between processingnodes such that data for all pairs of particles separated by less thanan interaction radius are available on at least one processing node. Twoor more types of computations are performed for pairs of particlesavailable at a processing node.

Aspects can include one or more of the following features.

The two or more types of computations include at least two of:electrostatic interaction, bonded force, particle migration, chargespreading, and force spreading computation.

The two or more types of computations includes electrostatic interactioncomputation.

In another aspect, in general, a method for performing computationsinvolving sets of items includes associating each of a multiple itemswith a set of groups to which the item belongs. Multiple sets of theitems are identified as candidates for performing a computationassociated with each set. Sets are excluded from the computationaccording to the group membership of the items in the set.

Aspects may include one or more of the following.

Excluding the sets includes excluding a set if multiple of its items areassociated with a common group.

Excluding the sets includes excluding a set if all of its items areassociated with a common group.

The items are associated with bodies in a simulation system.

The bodies form linked sets of bodies and at least some of the groupsare each associated with a linked set of bodies.

Each of the items associated with the bodies in the set of bodies isassociated with the group associated with the linked set.

The linked sets of bodies include bonded sets of bodies.

Identifying the plurality of sets of the items as candidates includesidentifying the sets such that the associated bodies fall within aprescribed interaction radius.

In another aspect, in general, a software product is adapted to performall the steps of any of the above methods. The software product caninclude instructions for performing the steps embodied on acomputer-readable medium.

In another aspect, in general, a parallel processing system is appliedto computing particle interactions. The system includes multiplecomputation nodes arranged according to a geometric partitioning of asimulation volume. Each of the nodes includes a storage for particledata for particles in a region of the simulation volume. The system alsoincludes a communication system including links interconnecting thecomputations nodes. Each of the nodes includes a processor subsystem,the processor subsystems of the computation nodes together, in adistributed manner, coordinate computation of the particle interactions.

Aspects can include one or more of the following.

The communication system forms a toroidal mesh.

The communication system includes separate components each linking adifferent neighborhood of computation nodes.

Each separate component of the communication system includes apoint-to-point link between a two of the computation nodes.

Sending at least some of the particle data to the other of thecomputation nodes includes passing that data via other computation nodesover multiple of the separate components of the communication system.

Each of the computation nodes further includes a communication subsystemof the communication system.

The communication subsystem is configurable to pass particle data fromone computation node to another computation node without intervention ofthe processor subsystem

Each node is configured to perform one or more of:

(i) send particle data to other of the computation nodes requiring thatdata for computation,

(ii) accept particle data sent from other of the computation nodes,

(iii) perform computations using the accepted particle data producingfirst results,

(iv) send the first results to other of the computation nodes,

(v) accept the first results send from other of the computation nodes,and

(vi) perform further computations using the accepted first resultsproducing second results.

The system is configured to perform all steps (i) through (vi).

The system is configured to perform some or all of steps (i) through(vi) in sequence.

The system is configured to repeat steps (i) through (vi) in eachiteration of a simulation.

The communication system is configured to accept the particle data froma computation node, and to distribute that particle data to aneighborhood of other of the computation nodes.

The communication system includes storage for configuring theneighborhood for distributing the particle data.

Each of the computation nodes further includes a communication subsystemof the communication system, the communication subsystem beingconfigurable to pass particle data from one computation node to anothercomputation node without intervention of the processor subsystem.

Each of the computation nodes further includes a computation subsystemfor performing the computations using the accepted particle dataproducing the first results.

The computation subsystem is further for sending the first results toother of the computation nodes, without intervention of the processingsubsystem at that node.

Each of the computation nodes further includes a memory subsystem foraccepting the first results send from other of the computation nodes,without intervention of the processing subsystem.

The memory subsystem is further for accumulating the accepted firstresults for particular of the particles.

The memory subsystem is further for providing data based on the acceptedfirst results to the processing subsystem.

The processor subsystem is further for performing further computationsusing the accepted first results and producing the second results.

The particle data includes particle location data.

The first results include particle force data.

The second results include particle location data.

In another aspect, in general, a processing node for use in a parallelprocessing system includes a group of subsystems. This group ofsubsystems includes a processor subsystem, a computation subsystem, anda memory subsystem. The system also includes a communication subsystemlinking the subsystems of the group of subsystems and providing at leastpart of a link between subsystems of the node and subsystems of otherprocessing nodes of the processing system. The computation subsystemincludes circuitry for performing pairwise computations between dataelements each associated with a spatial location.

Aspects can include one or more of the following features.

The processor subsystem includes is programmable for substantiallygeneral computations.

The processor subsystem includes a plurality of independentlyprogrammable processors.

The communication subsystem includes a set of interface units eachlinked to a subsystem of the group of subsystems, the communicationsubsystem also including one or more interface units for linking withinterface units of other processing nodes.

The communication subsystem includes a communication ring linking thesubsystems of the group of subsystems.

The communication system provides addressable message passing betweensubsystems on the same and on different processing nodes.

The computation subsystem is adaptable for computations involving pairsof particles in a particle simulation system.

The computation subsystem is adaptable for performing computations eachinvolving a particle and a grid location in a particle simulationsystem.

The group of subsystems and the communication subsystem are integratedon a single integrated circuit.

The memory subsystem includes an interface to a storage device externalto the integrated circuit.

The communication subsystem includes an interface for communicating withthe communication subsystems on other integrated circuits.

In another aspect, in general, a computation system includes an array ofprocessing modules arranged into one or more serially interconnectedprocessing groups of the processing modules. Each of the processingmodule includes a storage for data elements and includes circuitry forperforming pairwise computations between data elements, each of thepairwise computations making use a data element from the storage of theprocessing module and a data element passing through the seriallyinterconnected processing modules.

Aspects can include one or more of the following.

The array of processing modules includes two of more seriallyinterconnect processing groups.

At least one of the groups of the processing modules includes a serialinterconnection of two or more of the processing modules.

The system further includes distribution modules for distributing dataelements among the processing groups.

The system further includes combination modules for combining results ofthe pairwise computations produced by each of the processing groups.

In another aspect, in general, a processing module includes a set ofmatching units, each unit being associated with a storage for adifferent subset of a first set of data elements. The processing modulealso includes an input for receiving a series of data elements in asecond set of data elements, the input being coupled to each of thematching units for passing the received data elements to the matchingunits. The module also includes a computation unit coupled to thematching units for receiving pairs of data elements selected by thematching units, for each pair, one data element coming from the firstset and one data element coming from the second set, and an output fortransmitting results produced by the computation unit.

Aspects can include one or more of the following features.

The module includes an output for transmitting the received series ofdata elements to another module.

The module includes an input for receiving results produced by othermodules.

The module includes a merging unit for combining the results produced bythe computation unit and the results received on the input for receivingresults.

The merging unit includes circuitry for merging results according to adata element of the first set that is associated with each of theresults.

In another aspect, in general, a method includes receiving a first setof data elements, distributing the first set of data elements to thestorages of an array of processing modules that are arranged into one ormore serially interconnected processing groups of the processingmodules, receiving a second set of data elements, and passing the dataelements through the serially interconnected processing groups. At eachof at least some of the processing modules, a pairwise computation isperformed involving a data element from the first set that is stored inthe storage of the processing module and a data element passing throughthe processing group, and a result of the pairwise computation is passedthrough the serially interconnected processing group. The method furtherincludes combining the results of the pairwise computations passed fromeach of the processing groups and transmitting the combined results.

Aspects can include one or more of the following.

The data elements of the first set and of the second set are associatedwith bodies in a multiple body simulation system.

The pairwise computation is associated with a force between a pair ofthe bodies.

The method further includes, at each of the at least some of theprocessing modules, selecting some but not all of the data elements dataelements passing through the processing group for the performing of thepairwise computations.

Selecting the data elements includes selecting the pairs of dataelements for the performing of the pairwise computations.

Selecting the pairs of data elements includes selecting the pairsaccording to separation between spatial locations associated with thedata elements.

Distributing the first set of data elements to the storages of the arrayincludes, at each of the modules, distributing data elements to aplurality of matching units, each matching unit receiving a differentsubset of the data elements.

The method further includes at each of the at least some of theprocessing modules at each of the matching units selecting some but notall of the data elements data elements passing through the processinggroup for the performing of the pairwise computations.

The method further includes at each of the at least some of theprocessing modules combining the outputs of the matching units, andpassing the combined outputs to a computation unit for executing thepairwise computation.

In another aspect, in general, a software product is adapted to performall the steps of any of the above methods. The software product caninclude instructions for performing the steps embodied on acomputer-readable medium.

Other aspects and features are apparent from the following descriptionand from the claims.

DESCRIPTION OF DRAWINGS

FIG. 1 is a table of asymptotic scaling for various decompositionmethods.

FIG. 2 is perspective view of regions for the Half-Shell (HS) method.

FIG. 3 is perspective view of regions for the Neutral Territory (NT)method.

FIG. 4 is perspective view of regions for the Snir's Hybrid (SH) method.

FIG. 5 is a view of regions in a simplified two-dimensional analog ofthe Half-Shell (HS) method.

FIG. 6 is a view of regions in a simplified two-dimensional analog ofthe Neutral Territory (NT) method.

FIG. 7 is a view of regions in a two-dimensional decomposition thatshares some characteristics with the Snir's Hybrid (SH) method.

FIG. 8 is a view of regions in a two-dimensional decomposition.

FIG. 9 is a view of regions in a two-dimensional decomposition.

FIG. 10 is a view of regions in a two-dimensional decomposition

FIG. 11 is a view of regions in a two-dimensional decomposition

FIG. 12 is a view of regions in a simplified two-dimensional analog ofthe Half-Shell (HS) method for commutative interactions.

FIG. 13 is a view of regions in a simplified two-dimensional analog ofthe Neutral Territory (NT) method for commutative interactions.

FIG. 14 is a view of regions in a two-dimensional decomposition thatshares some characteristics with the Snir's Hybrid (SH) method forcommutative interactions.

FIG. 15 is a view of regions in a two-dimensional decomposition forcommutative interactions.

FIG. 16 is a view of regions in a simplified two-dimensional analog ofthe Half-Shell (HS) method for commutative interactions with rounding.

FIG. 17 is a view of regions in a two-dimensional decomposition thatshares some characteristics with the Snir's Hybrid (SH) method forcommutative interactions with rounding.

FIG. 18 is a perspective view of regions for the (SNT) method withrounding.

FIG. 19 is a perspective view of regions for the “clouds” method.

FIG. 20 is a perspective view of regions for the “city” method.

FIG. 21 is a perspective view of regions for the “foam” method.

FIG. 22 is a view of regions of a two-dimensional multiple zone method.

FIG. 23 is a computation schedule for the regions illustrated in FIG.22.

FIG. 24 is a computation schedule for a multiple zone version of thedecomposition shown in FIG. 3.

FIG. 25 is an algorithm corresponding to the computation schedule inFIG. 24.

FIG. 26 is a graph of computational contributions versus degree ofparallelism.

FIG. 27 is a perspective view of regions for the “eighth shell (ES)”method.

FIG. 28 is a computation schedule for the decomposition shown in FIG.27.

FIG. 29 is a graph of import volume versus degree of parallelism.

FIG. 30 is a perspective view of regions for a “sliced half-shell”method.

FIG. 31 is a computation schedule for the decomposition shown in FIG.30.

FIG. 32 is a perspective view of regions for the (SNT) method withrounding.

FIG. 33 is a computation schedule for the decomposition shown in FIG.32.

FIG. 34 is a graph of import volume versus degree of parallelism.

FIG. 35 is a graph of import volume versus degree of parallelism.

FIG. 36 is a view of regions in a two-dimensional pseudo-randomassignment method.

FIG. 37 is a block diagram of a parallel implementation.

FIG. 38 is a flowchart.

FIG. 39 is a block diagram including elements of a processing node.

FIG. 40 is a diagram illustrating data flow in a processing iteration.

FIG. 41 is a functional block diagram of a particle-particle interactionmodule (PPIM).

FIG. 42 is a block diagram of a portion of a midrange subsystem.

FIG. 43 is a block diagram of an implementation of a PPIM.

FIG. 44 is a block diagram of a flexible subsystem.

FIG. 45 is a block diagram of a serialized implementation.

DESCRIPTION

Section headings below are provided as an aid to the reader, and shouldnot be construed to limit the description. For example, approachesdescribed in different sections should not be construed to beincompatible based on the section headings. A listing of the sectionheadings is as follows:

1 Overview 24 1.1 Molecular dynamics 24 1.2 Molecular force fields 261.3 Pairwise interactions 28 1.4 Mesh based computations 31 1.5 Documentorganization 33 2 NT, SH, and HS Methods 33 2.1 HS method 35 2.2 NTmethod 36 2.3 SH method 37 3 Generalized Decompositions 37 3.1 Theconvolution criterion 40 3.2 Assignment rules 42 3.2.1 Assignment rulesfor HS, NT and SH analog 42 methods 3.2.2 Midpoint assignment rule 433.2.3 Pseudo-random assignment rule 45 3.2.4 Computational loadbalancing 46 3.2.5 Translational invariance 50 3.3 Commutativeinteractions 52 3.4 The rounding criterion 53 3.5 Decompositions inhigher dimensions 55 4 Shared communication 59 5 Pair-lists andexclusion groups 63 6 Multiple Zone Methods 66 6.1 A Lower Bound forImport Volume 72 6.2 Optimization Priorities 73 7 Applications ofgeneralized decompositions 74 7.1 Eighth Shell Method 74 7.2 Sliced HalfShell Method 76 7.3 Multiple Zone NT and SNT methods 76 7.4 Foam Method78 7.5 Comparison of Communication Bandwidth 79 Requirements 7.6Multiple-particle interactions 80 8 Serialized implementation 82 9Cluster architecture 88 10 Parallel implementation 89 11 Applications107 12 Alternatives 108

1 Overview

Simulation of multiple-body interactions (often called “N-body”problems) is useful in a number of problem areas including celestialdynamics and computational chemistry. For example, biomolecular orelectrostatic particle interaction simulations complement experiments byproviding a uniquely detailed picture of the interaction betweenparticles in a system. A significant issue in simulations is thesimulation speed.

1.1 Molecular dynamics

One approach to molecular simulation is primarily physics-based,predicting structure based on a mathematical model of the physics of amolecular system. Proteins may be modeled at the atomic level, withindividual atoms or groups of atoms represented as point bodies in anN-body system. One of these methods is molecular dynamics (MD), in whichthe force on each particle is calculated and Newton's laws arenumerically integrated to predict the physical trajectory of each atomover time. An alternative is a class of Monte Carlo methods, whichstochastically sample the potential energy surface of a system.Physics-based methods can either be used to refine homology models or beapplied on their own, to determine protein structure from scratch. MDcan also be used to study the process of protein folding.

One factor in applying physics-based methods to protein structureprediction and other structure-based problems in computationalbiochemistry is that often these problems are not concerned with anysingle molecule or molecular event, but rather with the statisticalproperties of a very large collection of molecules. For instance, whenpredicting protein structure, we may not necessarily be interested inthe structure that one particular protein molecule may happen to foldinto, but rather the most probable structure for molecules of thatprotein, the structure that will be the norm among a large collection ofmolecules of that protein. When studying binding, the question may notbe whether a particular protein and ligand will bind in a particularinstance, but the concentration of bound ligands in the steady state,when a large number of proteins and ligands are put together.

A useful quantity to compute for these problems is the free energy of amolecular system. Free energy is not a property of a single state of asystem, but rather of an ensemble of states. In very general terms, thelower the free energy of an ensemble of states, the higher theprobability that a molecular system will be found in one of those statesat any given time. The free energy of an ensemble is computed by summingprobabilities over all states in the ensemble. The probability of astate is an exponential function of its potential energy that depends onthe temperature of the system (specifically, the calculation uses aBoltzmann distribution). In practical terms, this means that for proteinstructure prediction, it is insufficient to perform a single run of anMD simulation, stop at some point, and take the end conformation as thenative state. At a minimum, many conformations near the end of such asimulation should be sampled. More likely, multiple separate simulationsmay need to be run and the results compared.

Once the structure of a target protein has been obtained—through eithercomputational or experimental methods—computational methods can be usedto predict the protein's interactions with ligands, a task looselyreferred to as “the docking problem.”

Screening ligands for activity could in theory be done withphysics-based methods. Owing to the enormous computational burden ofthat approach, other methods have been developed. Generally, differentconfigurations of a protein and ligand are generated and a score iscalculated for each according to a heuristic scoring function. To reducethe number of configurations that must be tested, the protein is heldrigid or allowed a small degree of flexibility; the ligand is tested invarious poses and orientations at various points near the protein. Tofurther reduce the configuration space, it is necessary (or at leastvery helpful) to know the active site on the protein.

This heuristic screening may identify ligands that are likely to bind toa target protein, but it may not quantify that binding or predict otherproperties of the interaction, such as the concentration of the ligandnecessary to produce a certain effect. For this purpose, the bindingfree energy of the interaction, the free energy difference between theunbound and bound states of the system, can be used. Too small a bindingenergy can make the required dosage of a drug too high to be practical;too high an energy can cause toxicity or other side effects. Accuratecalculation of binding free energy is a significantly morecomputationally intensive task than simply determining whether twomolecules are likely to bind, and relies heavily on accurate structuralmodels of these interactions.

1.2 Molecular force fields

In physics-based methods for structure-based problems, the “inner loop”of these computational methods can be chosen to solve the followingproblem: given a molecular system, calculate either (a) the potentialenergy of the system as a whole or (b) the force on each particle due toits interactions with the rest of the system. (Note that the force in(b) is simply the three-dimensional vector gradient of (a), making thetwo variations of the problem similar computationally.) These forces andenergies are generally computed according to a particular type of modelcalled a molecular force field.

Force fields can model molecular systems at the atomic level, with atomsor groups of atoms typically represented as point bodies. Each pointbody has a set of associated parameters such as mass and charge (morespecifically a partial charge, so that the complicated electrondistribution caused by atomic bonding can be modeled with pointcharges). These parameters are determined at initialization according tothe atom's type and, in standard force fields, remain constantthroughout the simulation. Atom type is not as simple as an atomicnumber on the periodic table: an atom's parameters can depend upon whatparticles are near it and, in particular, what other atoms are bonded toit. For example, a hydrogen atom bonded to a nitrogen atom (an amine)may have a partial charge that is different than one bonded to an oxygenatom (an alcohol). There can easily be several types each of carbon,hydrogen, oxygen, and nitrogen. The set of atom types and the parametersused for them are one of the characteristics that define a particularforce field.

The interactions among atoms can be broken into several components. Thebonded terms model the interaction between atoms that are covalentlybonded. One such term effectively models a bond between two atoms as aharmonic oscillator, reflecting the tendency for two atoms to settle ata certain distance from each other, known as the bond length. Anotherterm reflects the tendency for two bonds to “bend” towards a certainangle. Yet another term takes into account the effect of torsion, the“twisting” of a bond owing to the relative angles it makes with twobonds on either side of it. Several other types of terms are possible,many of them cross-terms, which take into account the interactionbetween the basic types of terms. Each term contains one or moreparameters, and each parameter must have a value for each combination(pair, triplet, quartet, etc.) of atom types, leading to a profusion ofparameters from all of the terms and cross-terms.

The non-bonded terms in a force field model all-to-all interactionsamong atoms. One of the non-bonded terms calculates the electrostaticinteraction of charges attracting and repelling each other according toCoulomb's law. Another type of non-bonded term models the van der Waalsinteraction, a shorter-range interaction that comprises an attractivecomponent and a repulsive component. The repulsive component dominatesat very short distances of a few angstroms (abbreviated {acute over(Å)}, equal to 10⁻¹⁰ meters). In a rough sense, the repulsive force canbe thought of as keeping atoms from overlapping or crashing into oneanother. There are various ways to model the van der Waals potential.The attractive component is usually modeled as dropping off as theinverse sixth power of the distance between two particles, or 1/r⁶; therepulsive component can be modeled with a similar power function(usually 1/r¹², for computational convenience on general-purposeprocessors) or with an exponential. Such forces are in contrast theelectrostatic potential which drops off more slowly, as 1/r.

1.3 Pairwise Interactions

Simulations in many fields require computation of explicit pairwiseinteractions of particles. In addition to the molecular dynamics andMonte Carlo simulations in biochemistry discussed above, similarsimulations may be applied to materials science, N-body simulations inastrophysics and plasma physics, and particle-based hydrodynamic andequation of state simulations in fluid dynamics. To reduce thecomputational workload of such simulations, instead of computinginteractions between all pairs of particles, one approach is to computeinteractions only between pairs separated by less than some interactionradius, R. These interactions are referred to as “near interactions.”The remaining interactions (“distant interactions”) are either neglectedor handled by a less computationally expensive method. Even with thesavings associated with limiting the interaction radius, the nearinteractions still frequently dominate the computational cost. Methodsfor decomposing this computation for parallel and/or serial processingcan be useful in reducing the total time required and/or computation orcommunication required for these simulations.

Traditional methods for parallelization are described in a number ofreview papers, including Heffelfinger, “Parallel atomistic simulations,”Computer Physics Communications, vol. 128. pp. 219-237 (2000), andPlimpton, “Fast Parallel Algorithms for Short Range Molecular Dynamics,”Journal of Computational Physics, vol. 117. pp. 1-19 (March 1995). Theyinclude atom, force, and spatial decomposition methods. Spatialdecomposition methods can offer an advantage over atom and forcedecomposition methods because they can exploit spatial locality to onlyconsider pairs of nearby particles rather than all pairs in the system.Force decomposition methods, on the other hand, can offer an advantageover spatial and atom decomposition methods in that the communicationbandwidth required by each processor decreases as the number ofprocessors increases.

Two recently developed methods—Snir's hybrid (SH, “hybrid”) method,published as “A Note on N-Body Computations with Cutoffs,” by Marc Snir(Theory Comput. Systems 37, 295-318, 2004) and Shaw's Neutral Territory(NT) method, to be published as “A Fast, Scalable Method for theParallel Evaluation of Distance-Limited Pairwise Particle Interactions”by David E. Shaw, combine spatial and force decompositions. They have aproperty that the communications required decreases as the interactionradius decreases. They also have a property that the communicationrequired per node decreases as the number of processors increases.Referring to FIG. 1, a comparison of asymptotic scaling properties ofthe SH and NT methods with those of traditional methods shows that theSH and NT methods have the same scaling properties, which aresignificantly better than those of traditional methods. Specifically,both of these two methods have asymptotic complexity ofO(R^(3/2)p^(−1/2)), which grow more slowly with the number ofprocessors, p, than all but the force method, which does not exploitspatial locality and therefore does not improve as the interactionradius, R, is reduced.

Both the SH and NT methods can offer advantages over traditional methodsfor practical machine and simulation sizes. For instance, the NT methodcan be shown to always requires less communication that the SH method,but that the SH method can be optimized using features of the NT methodsuch that the difference was small. An optimized version of the SHmethod that makes use of features of the NT method is referred to belowas the SNT method.

The NT, SNT, and SH methods involve different import regions. That is,the set of particles imported by each processor differs from one methodto another. The methods, however, share a number of features. In boththe NT method and the SH method, each processor stores the particlesfalling in some rectangular box. In both methods, the processor thatcomputes the interaction between a pair of particles typically is notthe one on which either particle resides, but instead a third processorthat imports both of the particles of the pair. In both methods, eachprocessor imports particles from two interaction zones and considers allpairs of particles such that one particle is in one interaction zone andthe other particle is in the other interaction zone.

The description below addresses a general class of methods that includestraditional spatial decompositions, the SH method, the NT method, andthe SNT method as special cases. The communication requirements of avariety of specific instances of the general class are analyzed takinginto account both latency and bandwidth of the interprocessorcommunication network.

A criterion for selecting parameters of a particular method can dependon the properties of both the physical system to be simulated (e.g.,system dimensions, interaction radius) and the hardware available forthe simulation (e.g., number of nodes, network topology, and memorystructure including cache structure and sharing between nodes). For mostpractical parameter settings, the method that delivers the bestperformance according to such a criterion generally is not necessarilyone of the specific methods described above, but rather is a differentinstance of the general class of methods.

1.4 Mesh Based Computations

As introduced above, distant interactions between particles, forexample, between particles separated by more than the maximuminteraction radius, can be computed using less computationally expensivemethods. One such approach is based on the Ewald Method, as described indetail in copending applications U.S. application Ser. No. 11/171,619,U.S. application Ser. No. 11/171,634, International Application No.PCT/US2005/023184.

The essence of the Ewald method is to split the electrostatic potentialfield into two main parts, one of which can be computed efficiently inreal space (by calculating pairwise interactions) and the other of whichcan be handled efficiently in the Fourier domain using computations on aregular grid on the simulation region. (Note, however, that not allvariations on Ewald use the Fourier domain.)

Implementation of the part involving the Fourier domain calculations canbe divided into three phases. First, a discrete charge distribution

${\rho (r)} = {\sum\limits_{i = 1}^{N}{\sum\limits_{n}{q_{i}{\delta \left( {r - r_{i} - {nL}} \right)}}}}$

that is defined on the (infinite and periodic) continuousthree-dimensional space r, is mapped to a charge distributionρ^(S)(r_(m)) defined on the finite three-dimensional mesh according to

$\begin{matrix}{{\rho^{S}\left( r_{m} \right)} = {S \otimes \rho}} \\{= {\int_{- \infty}^{\infty}{{S\left( {r_{m} - r} \right)}{\rho (r)}\ {^{3}r}}}} \\{= {\sum\limits_{i = 1}^{N}{\sum\limits_{n}{q_{i}{S\left( {r_{m} - r_{i} - {nL}} \right)}}}}} \\{\approx {\sum\limits_{i = 1}^{N}{q_{i}{S\left( r_{im}^{\min} \right)}}}}\end{matrix}$

where S is a charge spreading function, the operator

denotes spatial convolution, m is an index for mesh points that liewithin the primary box, and the vector r_(im) ^(min) is the vector frommesh point m to the nearest image of particle i.

In a second phase, a potential function φ*(r_(m)) is computed at thegrid locations using a convolution operation on the charge distributionρ^(S)(r_(m)). This convolution can be implemented using Fast FourierTransform (FFT) techniques.

In the third phase, energy or force quantities defined at the particlelocations are found using a convolution approach that maps themesh-based potential function to those locations. In particular, theforce on the i^(th) particle can be computed as

F _(i) ^(σ) =q _(i) F ^(σ)(r _(i))

where F^(σ)=T′

φ* and

${T^{\prime}(r)} = {\frac{}{r}{{T(r)}.}}$

This third phase can be implemented as a sum

$F_{i}^{\sigma} = {q_{i}h^{3}{\sum\limits_{m{({r_{i},T})}}{{\varphi^{*}\left( r_{m} \right)}{{T^{\prime}\left( r_{im}^{\min} \right)}.}}}}$

Note that computationally, the first phase and the third phase involvesums involving pairs of locations, with one location of each pair beinga particle (i) location and the other location of the pair being alocation of a mesh point (m).

1.5 Document Organization

The remainder of this document is organized as follows. Sections 2through 7 generally relate to algorithms that are applicable tocomputing range-limited pairwise interactions between particles, or moregenerally, to performing range-limited computations associated withpairs of points in space, where the pair may correspond to a pair ofparticles in an N-body problem, or the pair may correspond to one gridpoint and one particle point, for example, as part of a charge spreadingor force interpolation method.

Sections 8 through 10 relate to software and hardware techniques, whichmay be applicable to the algorithms presented in Sections 2 through 7,or to an overall n-body simulation implementation. Note that thehardware techniques described, for example in Section 10, have aspectsthat are applicable to a much broader range of applications than n-bodysimulation.

Possible applications of the algorithms and software and hardwaretechniques are discussed briefly in Section 11.

2 NT, SH, and HS Methods

The recently presented NT and SH methods, as well as an example of aconventional spatial decomposition method, the “Half-Shell” or HSmethod, can be described in the following framework. We assume that thesystem of particles (bodies) being simulated resides within arectangular box called the global cell. In all three methods, we dividethe global cell into a regular, rectangular, three-dimensional grid ofsmaller boxes. We refer to these smaller boxes as homeboxes. In oneparallel implementation, we assign a one-to-one correspondence betweenhomeboxes and processors, and we refer interchangeably to a processorand its homebox. The methods are described below in the content of suchan implementation, with some alternative implementations being presentedafter this discussion.

Also, when we refer to a homebox or processor importing particles, thisshould be understood to include a processor receiving informationcharacterizing those particles, such as their location, physicalattributes, identities, etc. Each processor is responsible for updatingthe position of the particles that lie within its homebox at any pointin time. As a convention, the base coordinates of a given home box (orof any particle within that homebox) are defined as the coordinates ofthe low-coordinate corner of that homebox. The logical coordinates of ahomebox or particle are the integer mesh coordinates of the homebox. Thedimensions of each homebox are h_(x)×h_(y)×h_(z).

Each home box has an associated import region consisting of the regionof space from which the processor must import particles. Each home boximports all particles not already present within that homebox that areto be interacted in the home box. The NT, SH, and HS methods each usedifferent import regions.

The NT, SH, and HS methods each interact all particles in one volume ofspace with all particle in another volume of space. We refer to thesevolumes as “interaction zones” or simply “zones.” In each of thesemethods, the intersection of the two zones is the homebox.

We use the volume of the import region associated with each homebox as asimple model for (i.e., a proxy for or an estimate of) the communicationbandwidth required by pairwise particle interaction method. Such a proxyis most accurate for a fully connected network topology.

2.1 HS Method

In conventional decompositions, generally, two particles are interactedin the homebox in which one of them resides. In the HS methodspecifically, the location where a pair of particles are interactedfollows the following rule: the particles are interacted in the homeboxwith the smaller x base coordinate; if the two particles have the same xbase coordinate; the particles are interacted in the homebox with thesmaller y base coordinate; if they also have the same y base coordinate,they are interacted in the homebox with the smaller z base coordinate.If they also have the same base z base coordinates, they are in the samehomebox and the interaction is computed there. Interactions betweenparticles on the same homebox are referred to as “local interactions”while interactions between particles on different homeboxes are referredto as “remote interactions.”

Referring to FIG. 2, as a result of the rule, each homebox has an importregion that consists of half the points external to the homebox within adistance R of the boundary. Every particle in the import region 220interacts with every particle in the homebox 210. Every particle in thehomebox 210 also interacts with every other particle in the homebox 210.Equivalently, the two interaction zones are the homebox and the union ofthe homebox with the half shell.

The conventional HS method is guaranteed to consider all unique pairsseparated by less than R. The method also could interact some pairswhose separation exceeds R due to the finite volume of the homebox.These “excess interactions” can be eliminated (“filtered”) at the pairlevel as necessary. Local interactions will also need filtering to avoiddouble counting interactions between local particles. Such filtering canbe accomplished by a test within an inner loop of a nested computationaliteration of particles.

2.2 NT Method

In the NT method, two particles interact in a homebox that has the x andy base coordinates associated with one particle and the z basecoordinate associated with the other particle. More specifically, the xand y base coordinates of the homebox in which the particles interactare those of the particle with the smaller x base coordinate or, if bothparticles have the same x base coordinate, those of the particle withthe smaller y base coordinate. The z base coordinate of the homebox inwhich the particles interact is the z base coordinate of the otherparticle. If two particles have the same x and y base coordinates, theyare interacted on the homebox of the particle with the lower zcoordinate.

Referring to FIG. 3, the resulting import region of each homebox is theunion an “outer tower” 320 and an “outer plate” 330. The outer towercomprises all points that lie directly above or below the homebox 310within a distance R of the boundary. The outer plate 330 comprises halfthe points external to the homebox 310 that lie in its z-plane within adistance R of the boundary. The interaction zones are the “tower,” whichconsists of the union of the homebox 310 and the outer tower 320, andthe “plate,” which consists of the union of the homebox 310 and theouter plate 330.

By interacting the tower (310, 320) with the plate (310, 330), all nearinteractions are guaranteed to be computed at least once. However, toensure that each near interaction is computed only once, a filteringcriterion is used that is not required in the HS method. Namely, local(i.e., homebox) plate particles should only interact with local towerparticles and remote (i.e., non-homebox) particles in the upper half ofthe outer tower (or lower half, but not both).

The import requirements of the NT method can be reduced by anappropriate choice of the aspect ratio of the homebox appropriately. Anoptimum occurs when h_(x)=h_(y)>h_(z). That is, an optimum occurs whenthe homeboxes are “flattened” in the z dimension by an amount thatdepends on the interaction radius and the homebox volume, and generallycorresponds to the tow interaction zones having equal or approximatelyequal volumes.

2.3 SH Method

The SH method assumes a cubical homebox of side length h. We definer=┌R/h┐ and r %=┌√{square root over (r+1)}┐. Referring to FIG. 4, theimport region of the homebox 410 with logical coordinates (i, j, k)(including the homebox) consists of the union of two interaction zones.The first interaction zone is a contiguous region that we refer to asthe “base” 420 and consists of the set of homeboxes with logicalcoordinates (i+u, j, k −w₁) where uε[−r, r] and w₁ε[0r %, −1]. Thesecond interaction zone is a set of non-contiguous regions that we referto as the “comb” 430. It consists of the set of boxes with logicalcoordinates (i, j+v, k+w₂%), where vε[−r, r] and w₂ε[0, └r/r %┘]. Notethat both the comb 430 and the base 420 include the homebox 410.Interacting the comb 430 and base 420 ensures that all near interactionswill be computed at least once. As for the HS and NT methods, filteringcan be used to ensure that all near interactions are computed exactlyonce and that only near interactions are computed.

3 Generalized Decompositions

In a generalized class of methods, each processor imports particles fromtwo interaction zones and interacts particles from one zone withparticles from the other zone. The pairs of zones are chosen for theimplementation of the method to ensure that all near interactions arecomputed. Both the case where each particle interacts with otherparticles in a surrounding cube, and the case where each particleinteracts with other particles in a surrounding sphere are considered.Because a spherical region of radius R has roughly half the volume ofthe circumscribing cube of side length 2R, “rounding” allows reductionin communication bandwidth requirements.

For clarity of exposition, we initially consider a simplified problem inwhich:

-   -   A. The particles are restricted to the two-dimensional x-y        plane.    -   B. A pair of particles interact if and only if they are        separated by a distance less than R in the x-dimension and by a        distance less than R in the y-dimension (that is, the Euclidean        distance may be greater than R)    -   C. The homeboxes are square with side length h, and R is an        integer multiple of h. Likewise, the interaction zones consist        only of complete homeboxes (“voxelized method”).    -   D. The interactions are non-commutative. Thus, for a pair of        particles a and b, we must separately compute the influence of a        on b and the influence of b on a. (Note that the force between        two particles is usually treated as a commutative interaction as        by Newton's Third Law, the force due to an interaction on        particle a is equal and opposite the force on particle b.        Non-commutative interactions are also of practical interest; for        example, one might want to compute the influence of each        particle on nearby mesh points.)

These simplifications are progressively relaxed below after an initialdiscussion of the simplified problem.

The interaction neighborhood of a homebox is defined to be the smallestregion of space that is guaranteed to contain all particles with whichany arbitrary local particle (i.e., any particle in the homebox) couldinteract. In this simplified problem, the interaction neighborhood is asquare of side length 2R+h extending a distance R beyond the homebox inthe positive and negative x and y directions. Particles in the homeboxinfluence particles in the interaction neighborhood and vice versa.

Referring to FIG. 5, a simplified analog to the HS method (more of a“full” shell than a “half” shell approach in view of the non-commutativeassumption D above) involves each homebox 510 computing the influence onits own particles of all particles in its interaction neighborhood 530.This method requires that each homebox import all the particles in itsinteraction neighborhood, except of course for those that already residein the homebox.

Referring to FIG. 6, a simplified analog to the NT method involves eachhomebox 610 computing the influence of all particles in a row ofneighboring boxes 620 that span the interaction neighborhoodhorizontally on all particles in a column of neighboring boxes 630 spanthe interaction neighborhood vertically. Each homebox imports both theremote particles in the row of boxes 620 and the remote particles in thecolumn of boxes 630. The import requirements for this method aresignificantly lower than those of the HS analog, even though the totalnumber of interactions computed is the same.

Note that in both of these methods, each homebox computes the influenceof all particles in one region of space, the interaction zone consistingof homebox 510 and boxes 620, on all particles in another region ofspace, the interaction zone consisting of neighborhood 530 and boxes630. In these examples, the two zones of each method overlap. Theirintersection is the homebox 510, 610. The zones are translationallyinvariant in the sense that each homebox 510, 610 has the same relativespatial relationship with its two associated zones.

Other choices for the two interaction zones lead to alternativedecompositions. Referring to FIG. 7, a decomposition (referred to belowas the “SH analog”) that shares some characteristics with the SH methodhas one zone comprised of a set of equally spaced horizontal slabs 720,while the other zone that is comprised of a vertically oriented block730 whose height is equal to the spatial period of the slabs. Again,each homebox 710 imports the particles in the two zones (720, 730) andcomputes the influence of particles in the one zone 720 on particles inthe other zone 730.

Referring to FIG. 8, in another decomposition, one zone 820 comprises asquare area, a “brick,” centered on the homebox 810. The other zone 830is a “foam”-like structure that consists of boxes the size of thehomebox positioned such that their period in each dimension is equal tothe width of the zone 820. The homebox 810 computes the influence ofeach particle in the zone 820 on each particle in the zone 830.

3.1 The Convolution Criterion

Given a pair of translationally invariant zones associated with eachhomebox, verification that all required interactions are computed can beperformed using a convolution-based criterion. We define the receptionregion of a method as the region of space that influences the homebox.Specifically, the reception region is the smallest volume of space thatis guaranteed to contain all remote particles that influence localparticles if no additional filtering at the pair level is employed. Thecriterion is based on ensuring that the reception region of a homeboxincludes its entire interaction neighborhood.

To find the reception region, assume each homebox has twotranslationally invariant types of interaction zones associated with it,its “blue” zones and its “red” zone. The homebox computes the influenceof each particle in its red zones on each particle in its blue zones. Wefirst find the set of homeboxes whose associated blue zone includes thedesignated homebox. We then take the union of the red zones for allhomeboxes in this set to obtain the reception region of the designatedhomebox.

To compute the reception region graphically, we first reflect the bluezone about the designated homebox to show the set of homeboxes whoseblue zones include the designated homebox. In two dimensions, reflectionabout the designated homebox is equivalent to a 180° rotation about thedesignated homebox. We then translate the red zone such that it has thesame spatial relationship to each homebox in the reflected blue zonethat it initially had to the designated homebox. The reception region isthe union of these translated red zones.

This process is similar to a convolution operation. Technically, thereception region is the support of the discretized convolution of anindicator function over the reflected blue zone with an indicatorfunction over the red zone. For that reason, we term the criterion thatthe reception region includes the entire interaction region the“convolution criterion.” For the four decompositions illustrated inFIGS. 5 through 8, the reception region of the homebox is identical toits interaction neighborhood.

Referring to FIGS. 9-11, several other decompositions that also satisfythe convolution criterion. The decomposition shown in FIG. 9 is similarto that of FIG. 8, but here, the two interaction zones 920 and 930associated with a particular homebox 910 are not centered on thathomebox 910. Symmetry of the zones about the homebox is not required.The zone 920 extends up and to the right of the homebox 910. Note thatwhen reflected, the zone 920 extends down and to the left of thehomebox. The decomposition of FIG. 9 also differs from those of FIGS.5-8 in that the reception region of the homebox is larger than itsinteraction neighborhood and in that parts of the zone 930 lie outsidethe interaction neighborhood 950 (indicated by the solid outlinedsquare). If one were to eliminate the parts of the zone 930 that areoutside the interaction neighborhood 950, the reception region would nolonger cover the entire interaction neighborhood, so the decompositionwould no longer ensure that all required interactions were computed.

FIGS. 10 and 11 show two other decomposition schemes that also satisfythe convolution criterion. In FIG. 10, neither of the zones 1020 or 1030for the homebox 1010 is contiguous. Similarly, in FIG. 10, the zones1120 and 1130 of homebox 1130 are not contiguous.

3.2 Åssignment Rules

Decompositions can be described in terms of interaction zones associatedwith each homebox. An alternative description takes the form of anassignment rule, which specifies the processor that will compute theinfluence of one particle on another as a function of their locations oron the processors on which they initially reside. Such a description wasinformally used in describing some of the methods, including thepreviously presented HS and NT methods above.

Suppose that particles a and b are separated by a distance no greaterthan R in each dimension. We denote the spatial coordinates of particlesa and b by (x_(a), y_(a)), and (x_(b), y_(b)), respectively, and denotethe logical coordinates of the homebox of particle a, the homebox ofparticle b, and the homebox that computes the influence of a on b by(i_(a→b), j_(a→b)), (i_(b), j_(b)), and (i_(a→b), j_(a→b)),respectively. An assignment rule can be specified as (i_(a→b), j_(a→b))in terms of the homeboxes j_(a)) and (i_(b), j_(b)) that contain theparticles, or in terms of the locations (x_(a), y_(a)), and (x_(b),y_(b)) of the particles.

3.2.1 Åssignment Rules for HS, NT and SH Analog Methods

For the simplified analog of the HS method, which is illustrated in FIG.5, the assignment rule can be represented as (i_(a→b), j_(a→b))=(i_(b),j_(b)). For the simplified analog of the NT method, which is illustratedin FIG. 6, the assignment rule can be represented as (i_(a→b),j_(a→b))=(i_(a), i_(b)).

To specify the assignment rules for the decomposition schemesillustrated in FIGS. 7 and 8, we define the functions r and g, which areused in the specification of the assignment rule, for integer argumentsas:

${r(m)} = \left\{ {{\begin{matrix}{{1\mspace{14mu} {if}\mspace{14mu} {\exists{k \in {Z\mspace{14mu} {such}\mspace{14mu} {that}\mspace{14mu} m}}}} = {{3\; k} + 1}} \\{{0\mspace{14mu} {if}\mspace{14mu} {\exists{k \in {Z\mspace{14mu} {such}\mspace{14mu} {that}\mspace{14mu} m}}}} = {3\; k}} \\{{{- 1}\mspace{14mu} {if}\mspace{14mu} {\exists{k \in {Z\mspace{14mu} {such}\mspace{14mu} {that}\mspace{14mu} m}}}} = {{3\; k} - 1}}\end{matrix}{g\left( {m,n} \right)}} = {m + {r\left( {n - m} \right)}}} \right.$

The assignment rule for the decomposition scheme of FIG. 7, the SHanalog, can be represented as (i_(a→b), j_(a→b))=(i_(b), g(j_(b),j_(a))). The assignment rule for the foam-like scheme, which isillustrated in FIG. 8, can be represented as (i_(a→b),j_(a→b))=(g(i_(b), i_(a)), g(j_(b), j_(a))).

3.2.2 Midpoint Assignment Rule

The assignment rule can also be specified in terms of the locations ofthe particles a and b. For example, if mid(a, b) is used to denote apoint at the geometric midpoint between the coordinates of a and b(i.e., x_(mid(a,b))=(x_(a)+x_(b))/2), then the assignment rule can berepresented as (i_(a→b), j_(a→b))=(i_(mid(a,b)), j_(mid(a,b))). Moregenerally, a midpoint mapping m(a, b) from the coordinates of theparticles to another point (“a mid location”) in the space can be usedthat is not necessarily the geometric midpoint of the points, and amidpoint rule can be applied to three or more points with a suitabledefinition of a “midpoint.” For example, a midpoint of a set of pointsmay be defined as the center of the smallest sphere containing all thepoints. As another example, an average location (e.g., for eachcoordinate direction averaging the coordinates of the points in theset).

Given an assignment rule and the interaction radius R, the import regionnecessary to satisfy the rule can be determined based on the rule. Theassignment rule is a many-to-one mapping from pairs of particlelocations to logical homebox coordinates. For any homebox in the rangeof the mapping, the inverse in the domain of pairs of particle locationsis well defined. This inverse defines the import region for the homeboxthrough the union of the domain of each coordinate of the pairs in theinverse. The inverse can be specified either according to thecoordinates of the particles, or in terms of their logical (homebox)coordinates. The existence of an assignment rule that is defined on theentire domain of pairs of particle locations with the global cellguarantees that each pair of particles will interact on some processor,so application of the convolution criterion is not necessary in thiscase.

Note that in the case of the midpoint assignment rule using thegeometric midpoint mid(a,b) above, the import region extends at most adistance R/2 away from the homebox, because if one of the particles isfurther than R/2 away from the homebox and the particles aregeometrically separated by at most R, then the midpoint necessarilyfalls outside the homebox.

Referring to FIG. 36, a two-dimensional example uses an array of squarehomeboxes, each of which is h=8 Å on each side. A homebox 3610 is arepresentative homebox, with the solid lines delineating the homeboxes.In this example, the interaction radius is R=10 Å. As presented above,the interaction region 3620 extends R/2=5 Å around homebox 3610. Arepresentative pair of particles, a 3631 and b 3632 has a geometricmidpoint m 3633 that falls within the homebox 3610, and therefore thecomputation of the interaction between a and b is computed by theprocessor assigned to that homebox.

The midpoint method can also handle interactions that involve sets ofthree or more particles. Suppose that we wish to compute interactionsbetween all sets of m particles that fall within some sphere of radiusR/2 (if m=2, this is equivalent to the statement that we wish to computeinteractions between all pairs of particles separated by less than adistance R). In the midpoint method, the interaction between a set of mparticles is computed on the processor that contains their midpoint,defined, for example, as the center of the smallest sphere that containsall m particles. The import region of the midpoint method in them-particle case is identical to that for the two-particle case.

3.2.3 Pseudo-Random Assignment Rule

For a particular import region defined for each homebox, quitefrequently the necessary particle data for an interaction is imported toor already available on more than one node. For example, in the exampleshown in FIG. 36 with a homebox size of h=8 Å with a R=10 Å interactionradius, the data required for the interaction between particles a 3631and b 3632 is also available to the processor with homebox 3612, whichhas an import region 3622. In general, in this example, data for anypair of particles is available to nodes associated with up to 9=3²homeboxes (in a three dimensional version, 27=3³ homeboxes). For most ofthe interactions in this particular case, there are at least two nodesthat could perform the interaction and there are quite frequently many,many more.

An approach to assigning the computation to one of the nodes that hasthe data for a pair of particles does not necessarily use the midpointnode. Rather, in this approach, the computation is assigned in agenerally uniform pseudo-random manner from a set of possible nodes suchthat at least one node and preferably only one node performs thecalculation. For example, each processor that has the data for thoseparticles makes a pseudo-random determination using little if anycommunication between the nodes to determine whether it should performthe calculation. In the case where only one node can compute theinteraction, in this example, that node contains the midpoint of theparticle locations.

A specific way of implementing the pseudo-random determination involvesthe data for a particle that is distributed by its homebox including abitvector which indicates which nodes have copies. For a particular pairof particles, the logical AND of the bitvectors provides the indicationof the nodes that have copies of the data for both the particles. Thenumber of such nodes that has the data for both particle a and b isdefined to be the n_cc(a,b). Using a same random number generator, eachnode with copies of the particles generates the same pseudo-randomnumber in the range 0 to n_cc(a,b)−1. This random number is used toidentify the node that will perform the computation based on the ANDedbitvectors (e.g., identifying the bit position of the n_cc^(th) setbit). One way of implementing the pseudo-random number generator usesquantities associated with the specific particles as seeds. For example,each particle has a global index, index_(a), index_(b), respectively,and the pseudo-random number generator is implemented assymmetric_hash(index_(a), index_(b)) MOD rank(a, b), wheresymmetric_hash(n, m) can be a hash function implemented using any of anumber of well-known techniques

As another variant, of this approach, the pseudo-random function is notnecessarily uniform. For example, if a goal is the spread out thecomputations on available nodes that have the needed data, the randomfunction is biased to nodes that have relatively few pairs of particlesto choose from. This type of bias can be incorporated by providingshared load information to the nodes, or by providing information fromthe sending nodes related to the number of particles for which data isbeing sent to other nodes.

3.2.4 Computational Load Balancing

In an approach related to the pseudo-random assignment rule discussedabove, the fact that multiple nodes in general are available to performeach pairwise computation is used for load balancing, with the nodescooperating to determine which computations are performed at which nodein such a way that each computation is performed only once, and adesirable balancing of computational load is achieved. We refer to aclass of methods as “midpoint assured methods” where each processorimports the same data as would be imported in the midpoint methoddescribed above, but the assignment of interaction computations toprocessors is adjusted in such a way that the computation load is moreeven across the processors.

For expository simplicity, we first describe this method for aone-dimensional space where each “box” is simply an interval of lengthb. We also assume that R is no greater than b, where a set of mparticles interact if they all fall within some sphere of radius R/2. Wenumber the processors such that the home box of processor i is the ithbox from the left. The computation proceeds in the following manner:

First, each processor imports all remote particles within a distance R/2of its home box.

Each processor then finds its set of locally computable interactions,consisting of all combinations of m particles available to thatprocessor that fall within a sphere of radius R/2. Each processor idivides its set locally computable interactions into threenonoverlapping subsets: interactions that are also computable on itsleft neighbor (processor i−1), interactions that are also computable onits right neighbor (processor i+1), and interactions that are onlycomputable on processor i. We denote these subsets L_(i), R_(i), and C,respectively, and we denote their sizes by l_(i), r_(i), and c_(i).R_(i) and L_(i+1) contain exactly the same elements, namely allcombinations of m particles that are within a distance R/2 of theboundary between box i and box i+1, so r_(i)=l_(i+1).

Each processor i then sends the value of c_(i) to its left and rightneighbors. This is the only communication required by thisparallelization method beyond that required by the basic midpointmethod. Each processor i now has the values of c_(i−1) and c_(i+1) aswell as c_(i), l_(i), and r_(i).

Each processor i then determines which of the interactions in L_(i) andwhich of the interactions in R_(i) it will compute, and which of theseinteractions it will leave to its neighbors to compute. In order toensure that each interaction is in fact computed somewhere, neighboringprocessors must make consistent decisions about the division of laborbetween them. Processors i and i+1 both have access to the values ofc_(i), c_(i+1), and r_(i) (because r_(i)=l_(i+1)). Processor i willcompute the first k_(i) interactions in R_(i), while processor i+1 willcompute the last r_(i)−k_(i) interactions, where processors i and i+1independently compute identical values for k_(i) based on c_(i),c_(i+1), and r_(i). To avoid missing or duplicating computation of anyinteraction, processors i and i+1 agree on the order of the interactionsin the equivalent sets R_(i) and L_(i+1); such a consistent orderingmight be based on the positions of the particles involved in eachinteraction, with ties broken by other particle characteristics such ascharge. In the absence of prior statistical information about thespatial distribution of particles, we might compute k_(i) as

$k_{i} = {\max \left( {0,{\min \left( {r_{i},{{round}\left( {\frac{r_{i}}{2} + \frac{c_{i + 1} - c_{i}}{3}} \right)}} \right)}} \right)}$

That is, if c_(i+1) and c_(i) are identical, then we assume thatprocessors i and i+1 have a comparable load, and we therefore split theinteractions in R_(i) evenly between them. If c_(i+1)>c_(i), then weassign more than half the interactions in R_(i) to processor i, in anattempt to balance the load. The max, min, and round operations in theabove equation are necessitated by the fact that k_(i) can only take oninteger values between 0 and r_(i). In the absence of such restrictions,distributing interactions according to the formula

$k_{i} = {\frac{r_{i}}{2} + \frac{c_{i + 1} - c_{i}}{3}}$

for all i would result in the assignment of an identical number ofinteractions to processors i and i+1, if c_(i−1) and c_(i+2) were bothequal to the average of c_(i) and c_(i+1). This average represents areasonable estimate of c_(i−1) and c_(i+2) based on information to whichprocessors i and i+1 both have access. Better load balancing could beachieved if both processors had access to c_(i−1) and c_(i+2), or ifprocessors had some prior statistical information about the particledistribution (e.g., the average particle density).

Generalizations of this method to higher dimensions can take severalforms. A computationally efficient and easily implemented generalizationis available if we expand the import region slightly such that its outerboundary is a rectangular parallelepiped extending a distance R/2 beyondthe interaction box in the positive and negative directions along eachcoordinate axis. We describe this generalization in the two-dimensionalcase, where the outer boundary of the import region will be a rectangle(a square if the interaction box is a square). Assuming that the sidelengths of the interaction box are all larger than the import radius R,an individual interaction may be computable on four boxes that share acommon corner, on two boxes that share a common edge, or only on asingle box. Our strategy is first to assign each interaction that can becomputed by two or four boxes in neighboring columns to a unique column,and then to assign interactions that can be computed by boxes inneighboring rows to a unique row. We perform these assignments byrepeated application of a procedure similar to that used to split R_(i)in the one-dimensional case, with appropriate values substituted forr_(i), c_(i), and c_(i+1) in Equation 1.

More specifically, we assume that processor (i,j) is in the ith row andthe jth column. After importing particles in its import region,processor (i,j) divides its set of locally computable interactions intonine subsets, based on which other processors can compute them. Wedenote these subsets by S_(i,j) ^(a,b), where a and b can each take onthe values −1, 0, or 1. The value of a indicates whether theinteractions in S_(i,j) ^(a,b) can be computed by processor (i−1, j) (inwhich case a=−1), by processor (i+1,j) (in which case a=1), or neither(in which case a=0). The value of b indicates whether the interactionscan be computed by processor (i,j−1) (in which case b=−1), by processor(i, j+1) (in which case b=1), or neither (in which case b=0). The shapeof the import region implies that an interaction is in S_(i,j) ^(1,1)and only if it is computable in both processors (i,j) and (i+1,j+1)(this would not be the case if the import volume were restricted to thatof the midpoint method). Neighboring processors agree on their set ofshared elements; for example, S_(i,j) ^(1,1) contains the same elementsas S_(i+1,j+1) ^(−1,−1), S_(i+1,j) ^(−1,1), and S_(i,j+1) ^(1,−1). Wedenote the size of S_(i,j) ^(a,b) by s_(i,j) ^(a,b).

Each processor (i,j) sends the values s_(i,j) ^(1,0), s_(i,j) ^(0,0),and s_(i,j) ^(−1,0) to processors (i,j−1) and (i,j+1). For each a in{−1, 0, 1}, we assign the first k_(i,j) ^(a,1) interactions in S_(i,j)^(a,1) to processors in column j and the last s_(i,j) ^(a,1)−k_(i,j)^(a,1) to processors in column j+1, where k_(i,j) ^(a,1) is computed byEquation 1 with s_(i,j) ^(a,1) substituted for r_(i), s_(i,j) ^(a,0)substituted for c_(i), and S_(i,j+1) ^(a,0) substituted for c_(i+1).Processor (i, j+1) independently computes identical column assignmentsfor all interactions in S_(i,j) ^(a,1). Processors (i+1,j) and (i+1,j+1) independently compute identical column assignments for interactionsin S_(i,j) ^(1,1), and processors (i−1, j) and (i−1,j+1) independentlycompute identical column assignments for interactions in S_(i,j)^(−1,1). At this point, all interactions have been uniquely assigned toa column. Interactions are then distributed between processors withineach column using a procedure identical to that described previously forthe one-dimensional case.

One could formulate multi-dimensional midpoint-ensured methods thatwould require only the import volume of the basic midpoint method, atthe cost of a more complex algorithm for assigning interactions toprocessors. The midpoint-ensured method described above generalizes in astraightforward manner to the case where R is greater than one or moreof the box side lengths if one requires that an interaction between aset of particles always be computed either in the box containing theirmidpoint or in a neighboring box (i.e., a box that shares at least acorner with it). Other midpoint-ensured methods that remove thisrestriction can lead to further improvements in load balance.

3.2.5 Translational Invariance

The decomposition schemes we have described thus far are translationallyinvariant. That is, each homebox has the same spatial relationship withits associated interaction zones. Assignment rules allow the descriptionof decompositions that are not necessarily translationally invariant.

As an example of an assignment rule that is not translationallyinvariant, consider an assignment rule where i_(a→b)=i_(a), the leastsignificant bit of the binary representation of j_(a→b) is the leastsignificant bit of j_(a), and the other bits of j_(a→b) are identical tothose of j_(b). This decomposition is similar to that of the SH analogillustrated in FIG. 7, but the import region of homeboxes with an odd jcoordinate is different from that of homeboxes with an even jcoordinate.

Such non-translationally invariant assignment rules may enable betterperformance in tree-like networks than restriction to translationallyinvariant approached. In a tree-like network, communication betweencertain sets of neighboring homeboxes is less expensive or morepreferable than communication between other sets of neighboringhomeboxes. It is possible that translationally invariant schemes willgive optimal performance in translationally invariant networks,including toroidal networks and networks with all-to-all interconnectionschemes, while non-translationally-invariant schemes may be preferablewhen the network itself is not translationally invariant.

We now devote our attention primarily to translationally invariantdecompositions, recognizing that many of the approaches can be applieddirectly or with some modification to non-translationally invariantdecompositions.

A subset of translationally invariant schemes are separable in the sensethat the logical x coordinate of the interaction homebox depends only onthe logical x coordinates of the particles and similarly for theycoordinate. Most of the schemes discussed in this paper are separable,again recognizing that many of the approaches can be applied directly orwith some modification to the non-separable case as well.

3.3 Commutative Interactions

One of the assumptions (labeled D above) in the simplified problem thatis presented above was that for any pair of particles a and b, weseparately compute the influence of a on b and the influence of b on a.In many, if not most applications, these influences possess symmetrysuch that only one interaction needs to be computed. Even if theinteractions are not commutative, the computations may require the sameinputs, such as particle positions. We can take advantage of this toreduce communication requirements by computing both non-commutativeinteractions at the same homebox.

The convolution criterion is modified for commutative interactions toinclude not only the reception region but also a “transmission region.”The transmission region of a method is the smallest volume of space thatis guaranteed to contain all particles that are influenced by particlesin the homebox. We need to ensure that the union of a homebox'sreception region and transmission region includes its entire interactionneighborhood. The transmission region and reception region are closelyrelated. One can determine the transmission region with a proceduresimilar to that used to determine the reception region, but with theroles of the “red” zone and “blue” zone reversed. The result is that thetransmission region is simply the reception region reflected about thehomebox. The union of the two regions includes the entire interactionneighborhood if and only if the reception region includes a “symmetrichalf” of the interaction neighborhood. That is, the two regions includethe entire interaction neighborhood if for any point in the interactionneighborhood that does not fall within the reception region, the pointobtained by reflecting it about the homebox falls within the receptionregion.

If assignment rules are being used to specify the decomposition method,one way to generalize an assignment rule to take advantage ofinteraction symmetry is to order the particle pairs before applying theassignment rule. That is, a new assignment rule can be defined as theoriginal assignment rule applied to the result of a function that ordersthe two arguments in a canonical order. For example, the ordering couldbe such that the first particle in a pair has the minimal x-coordinatewith ties broken by y. Some assignment rules (such as the rule for theeighth shell method described later in this paper) require no additionalordering to utilize the commutative nature of the interactioncomputations.

FIGS. 12 and 15 illustrate modifications that take advantage ofcommutative interactions for the import regions of HS and foam-likeanalogs of FIGS. 5 and 8, respectively. In each case, we have roughlycut one of the two interaction zones in half. The reception region foreach scheme is enclosed within a dashed line, 1250 and 1550,respectively. In each case, the reception region includes at least halfthe interaction neighborhood and the union of the reception region andthe reflection of the reception region about the homebox covers theentire interaction neighborhood. Note that in these decompositions, ifwe simply interact all particles in each zone with all particles in allother zones, then some near interactions will be computed and someexcess interactions for pairs that are separated by a Euclidean distancegreater than R might be computed. While pair level filtering mechanismscan be used, for example, by introducing a test in an inner loop of acomputation iteration, an alternative method that avoids suchdouble-counting without necessarily requiring such a test is introducedbelow in Section 6.

3.4 The Rounding Criterion

As introduced above, the approaches above relate to computation ofinteractions between particles that are separated by a Euclideandistance less than R, the interaction radius. In the simplifiedtwo-dimensional problem, each particle therefore interacts withparticles falling within a circle of radius R rather than a square ofside length 2R, as set out in assumption B above. Likewise, theinteraction neighborhood of a homebox is a subregion of the squareregion considered above (as a consequence of assumption B) with thecorners of the subregion being “rounded” rather than extending to thelimits of the homeboxes that fall at least in part within theinteraction neighborhood. Use of such an interaction neighborhood thatreduces in volume through this rounding is taken advantage of to reducethe size of the import region and thereby reduce the amount ofcommunication needed to import information for the particles to beinteracted. This technique to reduce the size of the import region isreferred to as “rounding.”

The criterion that specifies the degree to which the boundaries of theimport region can be rounded can be expressed as follows: a point in aninteraction zone v may be discarded from a zone if it is further than Raway from the closest point in the set of zones with which v interacts.For the two-zone methods considered in this section, this means thatpoints in the red zone may be discarded if they are further than R awayfrom all points in the blue zone, and points in the blue zone may bediscarded if they are further than R away from all points in the redzone.

In FIGS. 16-17, rounded versions of the HS and SH analogs, respectively,show reduced import regions determined according to the roundingcriterion presented above as compared to corresponding FIGS. 12 and 14,which do not employ rounding. In FIGS. 16-17, the interactions regions1650 and 1750, respectively, are enclosed by the dashed line. Note thatin FIG. 17, the rounded reception region 1750 expends beyond theinteraction neighborhood 1755, which is enclosed by the dotted line.While rounding never increases import volume, it does not always reduceit. For example, the import regions of the particular instances of thefoam-like analog in FIG. 15 or the NT analog in FIG. 13 are not reducedin volume by rounding.

The rounding criterion also allows us to generalize the techniques ofthe previous section to arbitrary R (for example that are not anintegral multiple of the side dimension of a homebox) and is applicableto arbitrary home box aspect ratios. In such cases, the interactionzones may contain fractional homeboxes.

In a voxelized method the interaction zone consists of completehomeboxes (assumption C above), without making use of rounding, alllocal particles are tested for near interactions against the set ofparticles in that method's reception and transmission regions. When amethod is rounded such that zones contain fractional homeboxes, assumingthat entire homeboxes in the interaction neighborhood are imported, thisis no longer true. For example, a local particle near the corner of ahomebox may be tested for a near interaction against a different set ofparticles than a local particle in the opposite corner. The importingitself is preferably limited to the rounded regions, such that a homeboxin the import region provides only particles located in a roundedsubregion to the homebox performing the computation. In this way, fewerpairs of particles that are separated by more than R are considered bythe homebox performing the computation, and for some pairs of zonesbeing interacted, no test in the computation loop is required to avoidconsidering pairs of particles separated by more than R.

3.5 Decompositions in Higher Dimensions

The simplified techniques described above generalize to the higherdimensional cases (relaxing of assumption A above). Given a method whereeach homebox interacts particles in one translationally invariantinteraction zone with particles in another zone, we can use theconvolution criterion or an assignment rule to determine whether allparticle pairs within a distance R will be interacted and then round theinteractions zones with the rounding criterion. The three-dimensionalHS, SH, NT and SNT methods described above are specific instances ofthis general class of schemes.

A class of decompositions that we have found useful in practice usestranslationally invariant separable decompositions. Recall that adecomposition is termed “separable” if the logical x coordinate of theinteraction homebox depends only on the logical x coordinates of theparticles and similarly for the coordinates in other dimensions. Aseparable decomposition can be described by specifying a separatedecomposition for each dimension. As one example of such a separabledecomposition, one can use either of two decompositions for eachdimension:

-   -   A. A decomposition where one zone contains only a single        homebox, while the other contains a row of homeboxes spanning        the interaction neighborhood. This is the case for both        dimensions of the HS analog illustrated in FIG. 12 and of the NT        analog illustrated in FIG. 13, and for the horizontal dimension        of the SH analog illustrated in FIG. 14.    -   B. A decomposition where one zone consists of several contiguous        homeboxes, while the other consists of a set of non-contiguous        individual homeboxes spaced regularly across the interaction        neighborhood. This is the case for both dimensions of “foam”        analog illustrated in FIG. 15, and for the vertical dimension of        the SH analog illustrated in FIG. 14.

Decomposition A above can be viewed as a special case of thedecomposition B, but in terms of the properties of the resultingmethods, it is worthwhile to treat them separately.

We can construct one zone of such a method by selecting one of thefollowing four choices for each dimension:

-   -   a. All homeboxes    -   a′. A single homebox    -   b. A set of individual homeboxes spaced regularly across the        dimension    -   b′. A block of multiple contiguous homeboxes

The other zone will then be uniquely determined by the requirement thatit be complementary to the first zone. In each dimension, a and a′ arecomplementary, as are b and b′.

In three dimensions, there are 20 unique decompositions that fit thisdescription, assuming we take advantage of interaction commutativity byhalving one of the interaction zones along one dimension.

Without loss of generality, we require that the “red” zone be halvedalong the first dimension. This restricts the options for the firstdimension of the red zone to a and b. We can select any of the fourchoices for the second and third dimensions of the red zone. Becausethere is no distinction between these two dimensions, however, we imposean ordering a>a′>b>b′ on the choices and require that the choice for thesecond zone be no larger than the choice for the third zone. Thus we canselect the decompositions for the second and third zones in 4+3+2+1=10ways, so there are a total of 2×10=20 decompositions in this class. Notethat we have ignored details such as the exact positioning of each zonerelative to the home box when counting unique decompositions.

The HS and NT methods, which are illustrated in FIGS. 2 and 3,respectively, are two of these unique decompositions. The SNT isanother; as illustrated in FIG. 18 in which zones 1820 and 1830 of thehomebox 1810 are rounded. FIGS. 19-21 show three further methods in thisclass. Referring to FIG. 20, a “clouds” method has one zone 1920 of thehomebox 1910 that consists to a series of circular plates ofsuccessively smaller radius resulting from the rounding criterion and asecond zone 1930 that is continuous with the homebox. Referring to FIG.20, a “city” method has one zone 2020 of a homebox 2010 (hidden fromview) that consists of a set of vertical towers that are truncatedaccording the rounding criterion and a second zone 2030 that iscontinuous with the homebox 2010. Referring to FIG. 21, a “foam.” methodhas one zone 2120 that consist of a set of subzones that are regularlyarranged in a three-dimensional grid and truncated according to therounding criterion, and a second zone 2130 that is continuous with thehomebox 2110.

Rounding can be taken advantage of to different extents with the methodsillustrated in FIGS. 18-21. The NT and SNT methods illustrated in FIGS.3 and 18, respectively take advantage of two-dimensional rounding. Thatis, they round one or both interaction zones in order to eliminate partsof the zone that lie more than a distance R from the homebox whenprojected onto the plane. Such rounding guarantees that each particlewill be tested against all particles in a surrounding regionapproximately shaped as a cylinder of radius R and height 2R. Whenhomeboxes are very small relative to R, for example in the highparallelization limit, two-dimensional rounding reduces the size of therounded zone and of the reception region by a factor of π/4 compared tothe unrounded case. Other decompositions, including the HS, clouds, cityand foam methods, illustrated in FIGS. 2 and 19-21, respectively, allowthree-dimensional rounding. That is, they allow one to reduce one of theinteraction zones in order to eliminate most parts of the receptionregion lying more than a distance R from the homebox inthree-dimensional space. Such rounding guarantees that each particlewill interact with all particles in a region shaped approximately as asphere of radius R surrounding it, rather than some larger region. Thesize of the rounded zone and of the reception region can be reduced by afactor of π/6 compared to the unrounded case and by a factor of ⅔compared to the two-dimensional rounded case. Three dimensional roundingof a zone is possible when it uses decompositions a or b in all threedimensions.

As discussed below, it is desirable to balance the volume of the importzones in order to minimize import bandwidth requirements. This is moreeasily accomplished in systems with an even number of dimensions, wherewe can select two interaction zones with the same shape (e.g., FIG. 6).Even in this case, we typically halve one zone along one dimension dueto symmetry considerations which introduces a difference in volumebetween the zones which could be balanced via other means. An odd numberof dimensions introduces an inherent asymmetry between the two zones. Wecan still balance their volumes, for example, in one of two ways:

-   -   Use non-cubical homeboxes, making one of the zones larger than        it would have been otherwise and the other smaller. This is the        technique employed by the NT method.    -   In methods where one or both zones involve multiple        non-contiguous pieces (e.g., decomposition b above), choose the        spacing between the pieces appropriately. This is the technique        employed by the SH method.

These optimization strategies can be combined. We return to the topic ofoptimization later.

4 Shared Communication

The midpoint method described above is particularly well-suited forparallelization of molecular dynamics simulations because it allowsseveral different components of the computation to utilize the samecommunicated particle data, thereby reducing total communicationrequirements more than it would if each component were parallelizedseparately. It can be used to parallelize the evaluation of bonded forceterms, which commonly involve the interaction of up to four particles,as well as pairwise electrostatic and van der Waals terms. Moreover, itallows several components of the MD computation, including theevaluation of the bonded force terms, to rely on the same data that iscommunicated for the evaluation of electrostatic and van der Waalsforces between pairs of nearby atoms.

The total force on a particle in most biomolecular force fields can beexpressed as a sum of bonded force terms, which depend on the covalentbond structure of the molecules, and non-bonded force terms, whichcomprise electrostatic and van der Waals interactions between all pairsof particles in the system. Bonded force terms include bond lengthterms, involving two particles connected by a bond; bond angle terms,involving three particles, two of which are bonded to a third; anddihedral angle terms, involving four particles connected by three bonds.We denote by r₁ the maximum radius of a sphere required to enclose theparticles involved in any bonded term. In common biomolecular forcefields, r₁ is no larger than 4 Å.

The non-bonded force terms, on the other hand, involve all pairs ofparticles in the system and therefore represent a much largercomputational burden. Van der Waals forces fall off sufficiently quicklywith distance that they can typically be neglected for pairs ofparticles separated by more than some cutoff radius r₂, typically chosenbetween 9 and 12 Å. An ever-mounting body of evidence shows thatneglecting electrostatic interactions beyond a cutoff is inadequate forexplicit solvent simulations, so electrostatic forces are typicallycomputed by one of several efficient, approximate methods that takelong-range interactions into account without requiring the explicitinteraction of all pairs of particles. Examples of these methods includemesh-based Ewald methods such as Particle-Particle Particle Mesh (PPPM),Particle Mesh Ewald (PME), Lattice Gaussian Multigrid (LGM), andGaussian Split Ewald (GSE), which use the Fast Fourier Transform (FFT)or a multigrid solver to compute electrostatic potential on a mesh givena charge distribution on that mesh. These methods require that modifiedelectrostatic interactions be computed explicitly within some cutoffradius; we will assume that this cutoff radius is chosen to be identicalto the van der Waals cutoff radius r₂, as is typically the case. Thesemethods also require that charge be mapped from particles to nearby meshpoints before the Fourier-domain or multigrid computation, and thatforces on particles are computed from potentials at nearby mesh pointsafterwards. In the charge spreading computation, the charge on each meshpoint is determined as a sum over nearby particles, while in the forcespreading computation, the force on each particle is computed as a sumover nearby mesh points. We denote by r₃ the maximum distance betweenany particle and a mesh point that “interacts” with it in either chargespreading or force spreading. The value of r₃ depends on the parametersof the long-range electrostatics method employed, but is typicallyaround 5 Å.

When one parallelizes the computation of explicit pairwise electrostaticand van der Waals interactions using the midpoint method, each processormust import data for particles lying within a distance of r₂/2 from theboundary of its home box. Each processor computes interactions thatexert force on the particles in its import region, so once it hascompleted computation of pairwise interactions, it must export a forcefor each of these particles to the respective particle's home box.

If r₃<r₂/2, then no additional communication is required to supportcharge spreading and force spreading. The import requirements of theexplicit nonbonded force computation ensure that each processor willimport all remote particles within a distance r₂/2 of any mesh pointlying in the home box. Likewise, the export requirements ensure thateach processor will export forces to this set of remote particles (aremote particle is a particle that does not reside on the interactionbox). The force contribution on each remote particle due to mesh pointsin the home box can be combined into each of these forces beforeinter-chip communication. In MD runs using common biomolecular forcefields, r₃ is usually approximately equal to r₂/2. Parameters of themesh-based Ewald method employed can be tuned to guarantee that r₃<r₂/2;alternatively, the import region can be expanded to include allparticles within a distance r₃ of the home box boundary.

Likewise, if r₁<r₂/2—as is typically the case for MD simulations usingcommon biomolecular force fields—then no additional communication isrequired to support computation of the bonded terms. This computationcan also be parallelized using the midpoint method, which specifies thateach bonded term be computed by the processor that contains the midpointof the particles involved in that term. The import requirements of theexplicit nonbonded force computation ensure that data for each particlewill be available on any processor that needs it to compute a bondedforce term, while the export requirements ensure that that a forcecontribution will be exported back to the processor on which thatparticle resides.

In an MD simulation parallelized via a spatial decomposition, particlesmigrate from one home box to another as they move. As long as themaximum distance r₄ that a particle moves in a time step—typically under0.1 Å—is less than r₂/2, no extra communication will be required tomigrate the particles; their new positions will be communicated as partof the import data required by the midpoint method for explicitnon-bonded interactions (in certain cases data will need to be importedfrom more boxes than if particle migration had been performedseparately, because the data to be imported might reside on any boxincluding points within a distance r₂/2+r₄ of the interaction box,rather than only on boxes including points within a distance r₂/2 of theinteraction box). Note that this is not generally the case for the NT,HS, or ES methods, in which proximity of a particle to an interactionbox does not guarantee that the particle will be imported by thatinteraction box.

Consider a molecular dynamics simulation with a cubic global cellmeasuring 64 Åon a side, running on a system with 512 processors andpartitioned into an 8×8×8 mesh of boxes. Suppose that we use aninteraction radius of r₂=12 Åfor explicit pairwise non-bondedinteractions. If we use the midpoint method to parallelize theseinteractions, the import volume will be 5923 Å³; no additionalcommunication is required for charge spreading, force spreading,particle migration, or bonded force computation, provided that bondedforce computation is also parallelized using a midpoint method. Usingthe HS method to parallelize the explicit pairwise non-bondedinteractions, on the other hand requires an import volume of 11352 Å³.This import volume does not ensure, however, that all remote particlesnear a mesh point on a particular home box are imported onto that homebox. If r₃=4.5 Å, charge spreading will require an additional importvolume of 2164 Å³, and force spreading will require a comparable amountof additional data export. Traditional parallelization strategies forbonded forces, in which each term is computed either on a fixedprocessor or on the processor containing a specific particle, willrequire additional communication, as will particle migration.

We expect that all method with the import region of the midpoint methodallow communication to be shared in the same manner. Also, while amesh-based Ewald methods is presented above, the midpoint method alsoallows communication sharing between components of the computation forother long-range electrostatic methods. Essentially all of these methods(including the Fast Multipole Method and related tree code methods)require that some pairwise particle interactions between nearbyparticles be computed explicitly; at a minimum, the communicationrequired for this computation can be shared with the communicationrequired for bonded interactions.

5 Pair-Lists and Exclusion Groups

In approaches that involve identifying pairs of particles that are to beinteracted, computation can be arranged such that the interactioncomputations are performed at the time that the pairs of particles areidentified. As an alternative, approaches such as the NT approach areused to identify the pairs of particles to be interacted, but ratherthan performing the computations as the pairs are identified, a list ofthe pairs is generated. The list is then used to iterate through therequired computations. For example, the pair list is completelycalculated before any pairwise computations begin, or alternatively, apipelined approach is used in which another software or hardware modulereceives the list and computes the interactions while more pairs arebeing computed. In some hardware architectures, separation of thegeneration of the pairlist from the pairwise computations may increasecomputation efficiency, for example, by allowing simpler hardware or amore efficiently coded inner software loop to be used to compute thepairwise interactions.

Another aspect of computation of pairwise particle interactions, whetherusing pairlists or performing the computations was appropriate pairs areidentified, is that some particle pairs within the specified interactionradius must effectively be excluded from the computation. For example,in a molecular dynamics simulation, the computation of Van der Waals andnear electrostatic interactions (non-bonded interactions) for allparticles that are within a cutoff radius R and that are not covalentlybonded is performed. If a pair of particles with the cutoff radius arecovalently bonded, then a near electrostatic force term is not included.A pair of particles within radius R but that does not have a non-bondedinteraction is termed an excluded pair.

Approaches to dealing with exclusion of computations involving bondedpairs of particles include (1) including the electrostatic forcecontributions for all pairs of particles within the interaction radius,and later subtracting out (i.e., “correcting”) the force contributionsfor pairs that are covalently bonded, and (2) identifying the excludedpairs during the identification of pairs of particles with theinteraction radius to avoid introducing the electrostatic force terms inthe first place.

Neutral territory methods described above allow interactions between allpairs within R to be found and distributed efficiently, for example, ona parallel machine. However, in order to take into account exclusionsefficiently during this computation, a means to quickly detect whetheror not a pair of particles is excluded can increase the efficiency ofthe computation.

One solution is to create a Boolean matrix indexed by particle whoseentries indicate whether or not a pair is excluded—conceptually,is_excluded(i,j). However, even for a modest number of particles, directstorage of this matrix (or half the matrix as the matrix is transposesymmetric) is likely prohibitive. Since the number of exclusions usuallyscales proportional to the number of particles, the matrix is sparse andcould be stored in a compressed format.

Exclusion groups provide an alternative means for specifying a set ofexcluded pairs. Each particle has an associated value that indicates theexclusion group to which the particle is assigned. If two particlesbelong to the same exclusion group, they are considered an excluded pairand discarded during the matchmaking process of a neutral territorymethod.

This test is efficient and maps nicely onto both neutral territorymethod implementations and the physical properties of many problemsinvolving range limited pairwise interactions. For example, in moleculardynamics again, all the non-bonded interactions between particlesbelonging to the same water molecule are commonly excluded for example.If there is an exclusion group that corresponds to each water moleculethen, no additional handling of water exclusions is necessary.

To demonstrate their usage and assignment, consider a simulation with 12particles. These 12 particles represent a methane molecule (particles0,1,3,6,9), two water molecules (particles 2,4,5 and 7,8,10) and asodium ion (particle 11). All pairwise non-bonded interactions betweenparticles that belong to the same molecule are excluded.

Particles 0,1,3,6 and 9 could be assigned to exclusion group 2,particles 2,4 and 5 could be assigned to exclusion group 5, particles7,8,10 could be assigned to exclusion group 0 and particle 11 could beassigned to exclusion group 1. This demonstration also shows thatparticles and exclusion group assignments have no special orderingrequirements.

6 Multiple Zone Methods

In general, in the methods considered above, two overlapping zones areinteracted—the homebox/shell, tower/plate, base/comb, brick/foam or,generically, blue/red. In this section, we consider methods involvingmultiple zones, any pair of which may be selected to interact dependingon the particular method being used. In such a multiple zone method, theimport region and homebox are treated as the union of a set of severalpotentially overlapping zones. Particles in a zone are interacted withall particles in a subset of the other zones.

To demonstrate a multiple zone method for a simple two-dimensional case,consider the import region shown in FIG. 22. The homebox 2210 imports arectangle 2220 adjacent to the homebox extending R in +x direction, arectangle 2230 adjacent to the homebox 2210 extending R in the +ydirection and a quarter of a circle 2240 of radius R adjacent to bothrectangles 2220 and 2230. The homebox 2210 is treated as a distinctzone. Homebox 2210 particles are interacted with the particles in theeach of the other three zones (2220, 2230, and 2240), and particles inthe vertical rectangle 2230 are interacted with particles in thehorizontal rectangle 2220. Completing the specification, pairs ofparticles are filtered by interaction radius R and local interactions(which occur exclusively in the homebox-homebox zone interaction) are tobe filtered to avoid double counting.

Note that the filtering operation can be implemented in a number ofways. If the homeboxes are smaller than an excluded volume about theparticles such that there is at most only one particle per homebox, thenthere will be no local interactions to consider at all. If there aremany local interactions, they could be filtered by particle ID or othermeans. Alternatively, they could be handled by a special case loopconstruct. If it is desirable to treat the homebox like any other zoneand the number of particles per homebox is modest, a fix-up loop couldbe done to subtract off redundant interactions. Still anotherpossibility for certain kinds of near interactions is to treathomebox-homebox zone interactions like regular zone-zone interactionsbut scale the results by half.

The import volume of this method is always less than that of the HSanalog, which is illustrated in FIG. 16. When R is small relative to thehome box side length, the import volume is also less than that of the NTanalog, which is illustrated in FIG. 13.

The validity of a multiple zone method can be checked by eitherdetermining an assignment rule or by applying the convolution criterion.The convolution criterion can be generalized to multiple zones usinggeometric considerations as follows. Each zone-zone interactiongenerates a reception and transmission region. The union of all thezone-zone reception and transmission regions must cover the interactionneighborhood of the homebox.

This method illustrated in FIG. 22 corresponds to the followingassignment rule:

(i _(a)

_(b) ,j _(a)

_(b))=(min(i _(a) ,i _(b)),min(j _(a) , j _(b)))

Alternatively, the convolution criterion can be applied. The receptionand transmission regions of the interaction between the homebox 2210 andthe other zones 2220, 2230 and 2240 cover all but two quarter circleportions of the interaction neighborhood. The interaction of therectangular zones 2220 and 2230 covers the remainder.

Note that for any particular multiple zone method, the set ofcombinations (pairs) of zones for which particle interactions areconsidered is selected, for instance, to avoid double counting. Theorder or schedule in which to consider combinations of zones can also bespecified, for example, selected according to computations orcommunication considerations. The order of considering combinations ofzones is not important to insure that all near interactions arecomputed, as long as all the selected combinations are considered.However, the schedule can be selected to overlap communications withcomputation.

For commutative interactions, the specification of a schedule can bethought of as an upper triangular matrix whose size is the number ofzones in the method. The entries in this matrix are non-negativeintegers giving the order in which the zone-zone interactions arecomputed. A zero entry indicates that the zone-zone interaction is notcomputed at all.

A schedule for the method in FIG. 22 is shown in FIG. 23. A multiplezone method is specified according to the definitions of the zones, thezone interaction schedule, and any additional redundancy filteringmechanisms.

An optimal schedule is influenced by characteristics of the network andprocessor on which the method is implemented. As one approach todetermining schedules, and the approach generally used in thisdescription involves filling the schedule matrix by row and then bycolumn. This order allows the maximum number of zone-zone interactionsto be performed with zones already received as the computationprogresses. Having selected an order for the rows and columns of thematrix, a remaining degree of freedom is the ordering of the zones.While a selection of an optimal order may in general be hardwaredependent, in the schedules shown in FIG. 23 import zones thatparticipate in more zone-zone interactions are scheduled before thosethat participate in fewer zone-zone interactions.

The order of interactions that allows maximal overlap of computation andcommunication depends on details of the available computationalhardware. A procedure for determining a schedule is as follows. First,we determine the order in which the zones will be imported, requiringthat zone u be imported before zone v if zone u participates in morezone-zone interactions than zone v. We then number the zones accordingto the order in which they are imported, with zone 1 representing theinteraction box, zone 2 representing the first zone imported, zone 3representing the second zone imported, and so on. We set scheduleentries corresponding to pairs of zones that should not be interacted tozero. We then number the remaining entries in column-major order; thatis, the interaction in row i₁ and column j₁ will be computed before theinteraction in row i₂ and column j₂ if j₁<j₂, or if j₁=j₂ and i₁<i₂.This ensures that computation of the interactions between a pair ofzones that have already been imported never has to wait for thecomputation of an interaction involving a zone that has not yet beenimported.

Multiple zone methods can have a number of practical benefits. First,they can reduce bandwidth requirements relative to previously knownmethods. Increasing the number of zones allows more near interactions tobe computed for a given import volume, because interactions are computedbetween zones rather than within a zone. Increasing the number of zonesallows more potential pairs to be considered for a given import volumebecause a given particle could be tested against particles from morethan two zones. An additional effect is that multiple zone methods canexpose additional rounding opportunities.

Second, multiple zone methods can be used to simplify or even eliminatemany of the filtering criteria required to avoid redundant computationsand/or double counting. For example, in the two-zone NT method, which isillustrated in FIG. 3, without redundancy filtering, interactionsbetween particles whose x and y base coordinates are the same (i.e., arein the tower consisting of the home box 310 and zone 320) are computedtwice. However, the NT method can be reformulated as a multiple zonemethod in a manner that eliminates the need for this testing. Thisreformulation is given below.

Third, using more zones allows more opportunities to concurrentlyperform communication for later-scheduled zone combinations whileperforming earlier zone combinations, thereby “hiding” communicationdelays with computation. For example, zones could be assigned in theschedule based on latency to import data from that processor. The orderin which zone-zone interactions are performed would then allow someinteractions to be performed earlier while more remote zones areimported.

To demonstrate these advantages on a three-dimensional method, considerthe original NT method reformulated using four zones. The zones aredenoted the homebox (H) 310, the lower tower (L), which forms a portionof zone 320, the upper tower (U), which forms another portion of zone320, and the outer plate (P) 330. The zone interaction schedule is shownin FIG. 24. The schedule eliminates the homebox-tower interactionredundancies; in particular, the homebox is not interacted with thelower tower. Consequently, the rounding criterion applied to the lowertower indicates that the corner furthest from the outer plate can betrimmed, reducing communication bandwidth

Note that the lower tower is trimmed by the intersection of twoperpendicular cylinders of radius R; the impact on bandwidth is smallfor large R but significant for small R.

Additionally, interaction filtering is simplified as there is no longera need to filter interactions between the particles in homebox andparticles in the lower tower. Referring to FIG. 25, an algorithm forimplementing communication and communication corresponding to theschedule in FIG. 24 gives a communication sequence based on the scheduleappropriate for MPI-style cooperative message passing systems for aforce computation. The algorithm has partially overlapped both particleimport and force export. At the beginning of computation cycle (one timestep of the simulation), the data characterizing the particles are sentfrom the homebox to other homeboxes that will need that data forcomputations at those homeboxes (step 2501). The other homeboxessimilarly begin sending their data and the homebox begins receiving datafor zones P, U, and L (step 2502). As the other homeboxes compute forceson particles in the homebox, they start sending the computed forces tothe homebox, which begins receiving them (step 2503). The homeboxalready has all the data for particles in the homebox, so it can performthe homebox-homebox interactions without waiting for the completion ofany of the communication that is in progress (step 2504). After thehomebox ends the step of receiving data for particles in the plate (P)that was initiated in step 2502 (step 2505), the homebox computes thehomebox-plate (H-P) interactions (step 2506). After the homebox ends thestep of receiving data for particles in the upper tower (U) that wasinitiated in step 2502 (step 2507), the homebox computes thehomebox-upper tower (H-U) interactions (step 2508) and computes theplate-upper tower (P-U) interactions (step 2509). At this point, thehomebox begins sending the computed forces for particles in the uppertower (U) to their respective homeboxes (step 2510). After the homeboxends the step of receiving data for particles in the lower tower (L)that was initiated in step 2502 (step 2511), the homebox computes theplate-lower tower (P-L) interactions (step 2512). At this point, thehomebox begins sending the computed forces for particles in the plateand lower tower to their respective homeboxes (step 2513). Finally, thehomebox finishes receiving the forces computed for its particles formother homeboxes (step 2514).

Another way a multiple-zone schedule approach can be used is to break upeach of the types of interaction zones described above into a collectionof uniform voxels. The voxels can have the same shape as the homeboxes,but this is not necessary. The schedule is then specified according towhich pairs of voxels are to be interacted, achieving the same result asa schedule define on the larger interaction zones, with an addedadvantage that better rounding can be achieved by only scheduling pairsof voxels for which some pairs of particles may be within theinteraction radius.

6.1 A Lower Bound for Import Volume

A lower bound on the import volume (V_(import)) of a multiple zonemethod can be established by considering the number of pairwiseinteractions that need to be computed. This bound takes the form:

$V_{import} \geq {{\frac{V_{h}}{\left( {1 - \frac{1}{N_{zr}}} \right)}\left\lbrack {\sqrt{{\left( {1 - \frac{1}{N_{zr}}} \right)\frac{V_{{ir},{remote}}}{V_{h}}} + 1} - 1} \right\rbrack}\mspace{14mu} {for}\mspace{14mu} N_{zr}} > 1$${V_{import} \geq {\left( \frac{1}{2} \right)V_{{ir},{remote}}\mspace{14mu} {for}\mspace{14mu} N_{zr}}} = 1$

Above, N_(zr) is the number of zones that import remote particles, andV_(h) is the volume of a homebox. V_(ir,remote) is the expected volumeof the portion of the interaction region that resides on otherprocessors for a randomly selected particle. V_(ir,remote) depends on R,V_(h), and the homebox aspect ratio. The lower bound decreases as thenumber of zones increases, as V_(h) increases, or as R decreases.

The bound for the import volume of a method is met when the remoteimport zones are balanced in volume, the interaction schedule is densein the sense that every zone interacts with every other zone, and themethod is efficient in the sense that every pair considered is anon-redundant remote interaction. Geometrically, this is often notfeasible, particularly when the interaction radius is small compared tohomebox dimension. One can approximate finite homebox size effects byreplacing V_(ir,remote) in the above bounds with the volume of theremote portion of a homebox interaction neighborhood. For sphericalinteraction regions, this volume (V_(ir,remote,eff)) can be expressed as

$\frac{V_{{ir},{remote},{eff}}}{V_{h}} = {\frac{4{\pi\alpha}_{R}^{3}}{3} + {{\pi\alpha}_{R}^{2}\left( {\alpha_{x} + \alpha_{y} + \alpha_{z}} \right)} + {2\; {{\alpha_{R}\left( {\alpha_{x}^{- 1} + \alpha_{y}^{- 1} + \alpha_{z}^{- 1}} \right)}.}}}$

Above, the normalized homebox aspect ratios α_(x)=h_(x)/V_(h) ^(1/3),α_(y)=h_(y)/V_(h) ^(1/3), α_(z)=h_(z)/V_(h) ^(1/3) and theparallelization parameter α_(R)=R/V_(h) ^(1/3) are used. This effectiveinteraction region volume is exact in the limit of large α_(R) for anyN_(zr) and in the limit N_(zr)=1 for any α_(R).

6.2 Optimization Priorities

By symmetry, the volume of the effective remote interaction region isminimized for cubic homeboxes (α_(x)=α_(y)=α_(z)=1). This gives a lesstight approximate lower bound that is independent of homebox aspectratio:

$\frac{V_{import}}{V_{h}} \geq {\left( {1 - \frac{1}{N_{zr}}} \right)^{- 1}\left\{ {\sqrt{{\left( {1 - \frac{1}{N_{zr}}} \right)\left\lbrack {\frac{4{\pi\alpha}_{R}^{3}}{3} + {3{\pi\alpha}_{R}^{2}} + {6\alpha_{R}}} \right\rbrack} + 1} - 1} \right\}}$

This bound does not indicate that cubic homeboxes are optimal; it issimply a bound that is independent of homebox aspect ratio. There arethree terms involving α_(R) in the bound. These terms geometricallyoriginate from the 8 corners (the cubic term), the 12 edges (thequadratic term) and the 6 faces (the linear term) of the interactionneighborhood.

Referring to FIG. 26, the magnitude of the three terms is illustrated asa function of α_(R). The magnitudes of the three terms are normalized sothat the magnitude of the corner term is 1.0. In the low parallelismlimit, the face terms dominate the import volume, followed by edge termsand then corner terms. In this limit, a method should strive to minimizethe volume of imports required to cover the faces of the interactionneighborhood followed by the import of edges. As the parallelizationparameter increases though, these priorities change. The firsttransition point occurs at α_(R)>2/π˜0.64. After this transition, thepriorities are ordered edge, face, corner. The next transition occurs atα_(R)>(9/(2π))^(1/2)˜1.20; the priorities are then ordered edge, corner,face. The final transition occurs at α_(R)>2.25 when the prioritiesbecome ordered corner, edge, face.

These optimization priorities are approximate and meant to serve asguidelines in designing methods. Depending on the exact homebox aspectratio and the details of the system running the methods, the exacttransition points between various methods could be different.

7 Applications of Generalized Decompositions

In this section, specific instances of the generalized approach arepresented. While most methods are not proven to be optimal in anyparticular sense, in many cases the methods are believed to be optimalor very close to optimal in terms of bandwidth for their intendedregime. In most methods described, the only interaction filtering thatneeds to be applied is interaction radius and local interactionfiltering. That is, filtering is not generally required to avoidconsidering pairs of particles more than one in the system, other thanfor pairs of particles that come from the same homebox.

7.1 Eighth Shell Method

The import region for a HS method consists of 4 corner, 6 edge and 3face import sub-regions. A face subregion includes homeboxes that differin one dimension, an edge subregion differs in two dimensions, and acorner subregion differs in all three dimensions. Intuitively, thehalf-shell method might seem to be the best that could be achieved inthe low parallelism limit (α_(R)<0.64). This is not the case.

The eighth-shell (ES) method is the three-dimensional generalization ofthe two-dimensional method illustrated in FIG. 22. The ES method importregion consists of 7 remote zones of the homebox H 2710: 3 face zones(+x import subregion denoted FX 2720, and similarly for y (FY is hiddenin FIG. 22) and z, FZ 2722), 3 edge zones (+y+z import subregion denotedEX 2730, +x+z region denoted EY 2731, +x+y region denoted EZ 2732) and 1corner zone (+x+y+z import subregion denoted C 2740). Referring to FIG.23, an interaction schedule corresponds to the regions shown in FIG. 22.The eighth shell method is so named because asymptotically, its importregion consists of one-eighth the full shell surrounding the homebox.

The ES import region is smaller than the HS import by 3 edges and 3corners. The edge-face interactions (interactions 9, 11 and 13) coverthe interactions that are handled in the HS method by the interaction ofthe homebox with the 3 removed corners. Likewise, the face-faceinteractions (interactions 4, 6 and 7) cover interactions that werehandled by the interaction between the homebox and the 3 removed edges.Like its two-dimensional counterpart, the method can be described by anassignment rule

(i _(a)

_(b) ,j _(a)

_(b) ,k _(a)

_(b))=(min(i _(a) ,i _(b)),min(j _(a) ,j _(b)),min(k _(a) ,k _(b))).

The total import volume is:

$\frac{V_{{import},{es}}}{V_{h}} = {\frac{{\pi\alpha}_{R}^{3}}{6} + {\frac{{\pi\alpha}_{R}^{3}}{4}\left( {\alpha_{x} + \alpha_{y} + \alpha_{z}} \right)} + {\alpha_{R}\left( {\alpha_{x}^{- 1} + \alpha_{y}^{- 1} + \alpha_{z}^{- 1}} \right)}}$

This method may be an optimal bandwidth method in the low parallelismlimit. It is also highly symmetric which can have additional benefits onvarious network topologies. Referring to FIG. 29, the bandwidth of anumber of methods is compared to the standard of the half-shell (HS)method 2910. The eighth-shell 2914 method is compared against severalother methods and the approximate lower bound when using cubicalhomeboxes. The eighth-shell method conforms to the low parallelismoptimization priorities of minimizing face, edge, and corner regions.The optimization priorities indicate the origin of the contributions tothe import volume in terms of different geometric categories as afunction of the degree of parallelism. For example, at a low degree ofparallelism, face imports dominate the import region and therefore wouldbe preferable to address as a priority for further optimization.

7.2 Sliced Half Shell Method

A class of methods for the range 0.64<α_(R)<1.20 may be well matched tothe optimization priorities. Referring to FIG. 30, a Sliced Half Shell(SHS) consists of 4 corner import subregions 3010 (denoted as a singlezone C4) in a sliced hemisphere arrangement and 5 face imports 3020(denoted as the 5 zones N, S, E, W, L). Note that the S and W faceimports can be rounded further than the N and E face imports as they donot interact with the homebox 3030. A schedule corresponding to FIG. 30is given in FIG. 31.

7.3 Multiple Zone NT and SNT Methods

The schedule for the multiple zone NT method presented above, and shownin FIG. 24. Referring to FIG. 32, a multiple zone of the two-zone SNTmethod illustrated in FIG. 18, consists of an extended base extendedbase 3220, which is below the homebox 3210. The comb 3230 and theportion 3240 of the base that are level with the homebox form fourimport regions. The remainder of the comb 3250 forms remaining importregions. The multiple zone SNT method has a relatively complicatedschedule shown in FIG. 33.

As for the NT method, multiple zone version of the SNT method eliminatesredundant interactions and allows for some additional roundingopportunities. If the multiple zone SNT method is further generalized toinclude methods where there is no spacing between the bars of the comb,then the extended base disappears and the +x import subregion becomesunnecessary as it only interacts with the extended base. The remainingset of import subregions exactly correspond to the NT method. It can beshown that if one has the freedom to optimize the aspect ratio of ahomebox in addition to the spacing between the bars in the SNT method,the SNT method is optimized by turning it into an NT method.

At sufficiently high parallelization parameters when the homebox aspectratio is fixed, an SNT method with judiciously chosen periodicity willbe superior bandwidth wise to the NT method. This can be understood byconsidering the optimization priorities. Appropriate for the higherparallelization regimes, both methods do not import any cornersubregions. Further, both import the same number of edge-like areas.However, for a fixed aspect ratio, the NT method optimizes the number offace imports at the expense of balance of volume between the outer towerand outer plate. The SNT optimizes for balance of volume between theextended base and partial comb at the expense of the number of faceimports. Asymptotically, the face imports are negligible contributionsto the import volume so that for fixed aspect ratio, the SNT methodachieves lower import bandwidth than the NT method. Based on theoptimization priorities, the NT method is expected to be most viable inthe parallelization regime 1.20 <α_(R)<2.25, while the SNT method isexpected to become viable beyond α_(R)>2.25. Accordingly, the crossoveris at α_(R)˜2.7 for cubic homeboxes as can be seen in FIG. 34

7.4 Foam Method

The foam method is the three-dimensional generalization of the methodshown in FIG. 15. This variant of the foam method has the many roundingopportunities and imports a brick of homeboxes of dimension s×s×scentered in the x and y directions on the homebox whose top is levelwith the top of the homebox. The homebox also imports a foam ofhomeboxes with a periodicity of s×s×s. The bottom of the foam is levelwith the bottom of the homebox and the foam is rounded against the brickto form a hemispherical structure. (In the low parallelism limit, thebrick can be rounded as well but the method is not competitive in thatlimit.) Asymptotically, the import volume of this method is:

$V_{{import},{foam}} \sim {{s^{3}V_{h}} + \frac{2\pi \; R^{3}}{3\; s^{3}}}$

Optimizing the periodicity to minimize the import volume yields:

$s^{6} \sim \frac{2\pi \; R^{3}}{3\; V_{h}}$

Thus, the asymptotic import volume of an optimal foam method is:

${V_{{import},{foam}} \sim \sqrt{\frac{8\pi \; R^{3}V_{h}}{3}}} = \sqrt{\frac{8\pi \; \alpha_{R}^{3}V_{h}^{2}}{3}}$

The foam method achieves the exact lower bound asymptotically for amethod with two remote zones. Practically, this method is best matchedto situations in which the parallelization parameter is large. As can beseen in FIG. 34, foam only becomes competitive at parallelizationparameters in excess of ˜14. This is a consequence of the fact that asthe number of processors increase, the relative contribution of theoverhead of foam faces and foam corners to the import volume onlydeclines very slowly.

Implementations of foam methods can be coupled with zone scheduling on ahomebox-by-homebox level to “hide” import latencies from more distanthomeboxes. Since the number of homeboxes imported is large when foam ispractical, a zone interaction schedule is omitted for brevity.Conceptually, the key elements of the schedule are that the homebox isinteracted with both the foam and the brick, while the foam and brickare interacted with one another.

7.5 Comparison of Communication Bandwidth Requirements

FIGS. 34 and 35 summarize the import volumes of a number of methodsdescribed above over a range of parallelization parameters. For cubichomeboxes, the ES method is best among these methods for α_(R)<0.8. TheNT method is best between 0.8<α_(R)<2.7. The SNT method is best from2.7<α_(R)<˜14. Beyond, foam methods are increasingly competitive. Thesetransitions can be related to the number of processors used forparallelization. For example, consider a molecular dynamics simulationwith N=25,000 atoms and R=12 Å. Biological systems have a density ofatoms of roughly D=0.1/Å³. Assuming cubic homeboxes are being employedand that each atom is represented by a single particle, theparallelization parameter can be written in terms of these parameters:

$\alpha_{R} = {\frac{R}{V_{h}^{1/3}} = {\left. \frac{{RD}^{1/3}p^{1/3}}{N^{1/3}}\rightarrow p \right. = {\frac{\alpha_{R}^{3}N}{R^{3}D} \sim {145\alpha_{R}^{3}}}}}$

The graph in FIG. 34 indicates that when using below ˜75 processors, theeighth-shell method will minimize communication bandwidth (moderatesized clusters). Between ˜75 and ˜2,800 processors (large clusters), theNT method is best. Beyond ˜2,800 processors (for example, in the rangeof ultra parallel hardware such as in BlueGene or QCDOC), more elaboratemethods are preferable, beginning with the SNT method with a spacing of2. The foam methods begin to become preferable at the extreme half amillion processor level.

The relative import volumes of the various methods are different if thehomebox aspect ratio is allowed to float. FIG. 35 shows the relativeimport volumes when each method is allowed to optimize the homeboxaspect ratio, assuming a fixed global cell volume and number ofprocessors. In this case, the eighth-shell is still optimal forα_(R)<˜0.8, followed by the NT method, followed by the foam method atextreme parallelization levels. The floating homebox aspect ratio isparticularly important for fully interconnected networks and in themassive parallelism limit. In practice, the global cell aspect ratio isusually fixed by phenomena being simulated. The set of availablepotential homebox aspect ratios is then set by the network topologyand/or the number of ways the global cell can be partitioned among theprocessors. On fully interconnected networks though, the topology doesnot bias in favor of a particular homebox mesh. Similarly, if there area large number of processors available for the simulation, the mesh maybe partitioned very flexibly, especially if the exact number ofprocessors used is not fixed. As a result, on such hardware,communication schemes may be improved by optimizing homebox aspect ratioon a per method basis.

7.6 Multiple-Particle Interactions

In the description above, various approaches are described in thecontext of computing pairwise interactions. The approaches are alsoapplicable to computing interactions between sets of greater than twoparticles. For example, forces between atoms in a molecule can bearranged using these approaches. In the multiple zone approaches, forexample using the ES method, particular combinations of more than twozones can be scheduled in order to arrange such multiple particleinteractions.

For a midpoint method for interactions that involve three or moreparticles, computing the center of the smallest sphere containing mparticles is computationally nontrivial when m≧3. One option foravoiding this computation is to use midpoint-ensured methods that do notrequire computation of a midpoint. Another is to use an efficientapproximation for the midpoint. Two such approximations are:

-   -   a. For each of the coordinate dimensions, approximate the        coordinate of the midpoint as the arithmetic mean of the        coordinates of the particles.    -   b. For each of the coordinate dimensions, approximate the        coordinate of the midpoint as the arithmetic mean of the        coordinates of the particles.

When using either approximation, one needs to expand the import regionslightly to ensure that the box containing the approximate midpoint of aset of interacting particles will import all remote particles in thatset. In the absence of any restrictions on the spatial distribution ofparticles, use of approximation 1 requires that the import regioncontain all particles within a distance

$R\left( {1 - \frac{1}{m}} \right)$

of the interaction box boundary, implying a substantial increase inimport volume (if m−1 particles are collocated inside the interactionbox at a distance

$\frac{R}{m}$

from the boundary and the mth point is located outside the interactionbox at a distance just under

$R\left( {1 - \frac{1}{m}} \right)$

from the boundary, then the midpoint will fall within the boundary). InMD simulations using common biomolecular force fields, however, asmaller increase in import volume may be sufficient, because the spatialarrangement of the particles involved in a bonded term is not arbitrary;for example, any pair of atoms must maintain some minimal distance fromone another under normal conditions.

A sufficient import volume for use of approximation 2 is that enclosedby a rectangular parallelepiped extending a distance R/2 beyond theinteraction box in the positive and negative directions along eachcoordinate axis. One can show that a somewhat smaller volume suffices,one that includes all remote particles within a rectangularparallelepiped extending a distance R/4 beyond the interaction box ineach dimension as well as all particles within a distance R/4 of thatparallelepiped. In three or fewer dimensions, for m≧3, this importvolume is always smaller than that required by approximation 1 in theabsence of restrictions on the spatial distribution of particles.

8 Serialized Implementation

One approach to computation of particle interactions, which can includeone or more of the algorithmic techniques described above, make use of aserialized software implementation. In such an implementation, aprocessing node performs computations for homeboxes in a serializedmanner. For example, for each homebox, a multiple zone method isimplemented in which interactions between pairs of particles inrespective pairs of zones are computed by the processor. In one versionof such a serialized implementation, a single processing node performsall the computations for all the homeboxes in a serialized manner. Inanother version, a single processing node of a set of multipleprocessing nodes is responsible for a set of homeboxes, for example,that together define a region of space (e.g., a cube). The processingnode may have a number of processors that share at least some commonmemory. For example, each of the processors at a node may have a privatecache memory and access a shared memory that is local to the node as awhole. There may be further level of memory, for example, shared betweenprocessing nodes, but that is not directly significant to the discussionbelow.

Referring to FIG. 45, in one version of a serialized implementation, aprocessor 4540 uses a local memory 4550. For example, the local memory4550 may be a fast cache memory that is associated with the processor4540, and which provides local copies of data in a global memory 4560.The global memory 4560 holds the particle data 4562 for the entireglobal cell being simulated by the system.

One aspect of local or cache memories is that they are typically fasterthan larger memories, but are limited in size. Some arrangements of thecomputational iterations may result in relatively larger numbers ofmemory access to the global memory 4560 than others. To the extent thatthe number of global memory accesses is a significant factor thatdetermines the overall time required for the overall computation, sucharrangements of the iterations may require relatively more overallcomputation time.

There are speed advantages to maintaining all or most of the datarequired for a computation in the local memory 4550 if that data can beused multiple times. This reduces the overall memory latency in that aninitial fetch of the data may be relatively slow, but subsequentaccesses to the data that can use the cached copy of the data arerelatively fast.

One way of making effective use of limited size caches is to “cacheblock” the near interactions. The term “cache block” is used as ananalogy to cache blocking techniques used to accelerate matrix-matrixmultiplication. Naïve implementations of large matrix-matrixmultiplications require O(N³) data transferred to and from main memory(bandwidth) when the entire matrices being multiplied do not fit in thecache memory. Generally, this results from the entries in the matrixbeing repeatedly flushed and then reloaded into the cache as thecomputation continues. However, by breaking each matrix into acollection of “cache blocks” and determining an appropriate schedulecache blocks are loaded, processed and stored, matrix-matrixmultiplication can be structured to have O(N²) bandwidth.

In computation of pairwise interactions, in a worst-case use of thelocal memory, computation of every interaction between a pair ofparticles requires accessing the global memory 4560 for at least one ofthe particles. For example, this may result if the order in whichparticles are interacted is not performed with regard to theirlocations, or if a method such as the half-shell method is used but theparticle data for an entire half shell of data does not fit in thecache. In such a worst-case situation, the number of such global memoryaccesses is O((DV)(DR³))=O(D² V R³), where V is the volume beingsimulates, D is the density of particles, and R is the interactionradius. That is, DV is the total number of particles, and the averagenumber of particles in with the interaction radius of a particle isproportional to DR³. On average, for every load data for a particle intothe cache, essentially only a single pairwise computation is performed.

In a cache block arrangement of the computation of pairwise interactionsthe number of global memory accesses can be reduced to O(D V^(1/2)R^(3/2) N_(v) ^(1/2)), where N_(v) is the number of “voxels” used.Specifically, the volume V being simulated is partitioned into N_(v)voxels, in general, each being the same shape. In general, each voxel isanalogous to a homebox in the decomposition approaches described above.

Referring to FIG. 45, the general approach to arranging the computationof the pairwise interactions involves loading the particle data for animport region 4570 and its associated homebox (i.e., the interactionzones associated with the homebox) into an interaction region dataportion 4552 of the local memory 4550. The size of the import region,which depends on the volume and geometry of voxels, is chosen such thatthe data for the particles in the interaction region fits in the localmemory 4550. Once this data is loaded into the local memory, any pairsof interactions between a particle in the interaction region can becomputed without necessarily accessing the global memory.

The saving in access to the global memory can generally be understood bythe number of times the data for each particle is used in a pairwisecomputation. In the case of two interaction zone, if the local memory issufficiently large to hold the particle data for 2M particles (M fromeach of equal volume interaction zones), then up to M² interactions canbe computed. That is, there are M/2 pairwise computations for everyparticle loaded into the cache, resulting in a M/2 speedup with respectto memory access time using a naïve worst-case method.

As one example of a type of interaction region, an NT approach, asdescribed in Section 2.2 and illustrated in FIG. 3, is used. Theinteraction region 4570 is partitioned into voxels (voxel 4564 beingindicated as a representative voxel). The local memory 4550 alsoincludes a schedule that identifies the pairs of voxels that are to beinteracted. For example, each voxel in the tower portion of theinteraction region 4570 is interacted with each voxel in the plateportion. For each pair of voxels that are interacted, all the pairs ofparticles, one particle from each voxel, can be considered.

In other examples, the entire import region associated with a homeboxdoes not necessarily fit in the local memory. However, if the importregion is divided into multiple zones such that pairs of the multiplezones fit in the local memory, they can be loaded in the local memoryaccording to a multiple-zone schedule and all the pairs of particlesinteracted while the data remains in the local memory. In otherexamples, subsets of zones are loaded into the local memory and allscheduled combinations of pairs of the loaded zones are interacted whilethe data for those zones remains in the local memory. In these examples,generally, an M/2 speedup is also achieved to the extent that 2Mparticles are loaded and M² interactions are computed.

For an optimal choice of height of voxel (i.e. a square cross-sectiontower, and a relatively thin plate), the volume of the import region isO(R^(3/2)) rather than O(R²) as with cubic voxels. For the NT method,the geometry of the voxels can be chose by optimizing the height of thevoxel (the thickness of the plate) such that bandwidth scales as thetotal number of particles times the square root of the expected numberof particles in an interaction region. Often, there are 200-400particles expected in an interaction region so this square rootreduction can be significant.

Various architectures of local memory can be used with these approaches.In one example, the local memory is a cache memory that uses the sameaddress space as the global memory, and a cache controller maintainscopies of the most recently accessed data from the global memory in thecache. As another example, a non-uniform memory architecture can be usedin which the local memory uses a distinct address space than the globalmemory, and the processor explicitly copies the particle data from theglobal memory to the local memory as needed.

The approach can be extended to a multiple level cache structure inwhich a first level cache may have space to hold two subzones, which areread from a second level cache that holds more data, for example, theentire import region for a homebox.

The loading of data into the local memory may be aided by aspecial-purpose memory controller. For example, loading the data fromone voxel can use a regular and generally relatively large memorystructure that is performed using hardware external to the processor(e.g., special purpose FGPAs).

A multiple processor implementation can also be used in which eachprocessor has its own local memory, and all the local memories arecoupled to a shared global memory. The schedule of voxel pairs tointeract is then split among the processors so that together all therequired interactions are performed.

Between interactions, the particle data is reassigned to the voxels inwhich the are located (as a result of motion of the particles caused bythe forces computed in the previous iteration). This partitioning of thedata is performed using an efficient O(N) algorithm, where N is thenumber of particles. During the partitioning process, a data structure(an array) is built that allows efficient access to strips of adjacentvoxels along one or the orthogonal axes are needed when loading aninteraction zone into the local memory.

The order of voxels processed is chosen so that in addition to reducedmemory bandwidth, particles loaded into the cache blocks for theinteraction zones for one voxel will substantially overlap the particlesfor the interaction zones of the next voxel. This further acceleratesthe operations by increasing temporal hardware cache locality.

As described above, the computation involves computing interactionsbetween pairs of particles loaded into the local memory in aschedule-based loop. One alternative it to defer computation of theinteractions in this loop and rather computing a list of pairs ofindexes of the particles to be interacted. Then this list is processedin a second loop to actually compute the interactions. This second loopmay be accelerated using special-purpose hardware. Because the list ofpairs is already computed, such special-purpose hardware may berelatively simple.

Non-commutative (e.g., grid-particle) interactions can be organized sothat the zones and voxel sequencing is such that multiple processors canwrite without conflict to the same output arrays. For example, incomputations where the output is a charge density array, this caneliminate the need to use one charge array per processor and the need toreduce these partial charge arrays into a single output array.

A specific instance of a highly optimized cache-blocking method is thedual of the NT method described above and has four cache block zones.The first zone is a single homebox voxel. The next zone is a “tower” ofvoxels. It consists of all non-homebox voxels along an axis-alignedvoxel ray that starts on the homebox voxel and that contain at least onepoint within R of the closest point in the homebox voxel. The last twozones form an approximately circular plate of radius ˜R. These two zonesconsist of all non-homebox voxels that contain at least one point withinR of the closest point in the homebox voxel in the plane perpendicularto the tower that contains the homebox voxel. These two zones are splitevenly into a “yin” zone and a “yang” zone. These were so named as themost convenient division of the plate results in symmetric zones thatfit together like a yin-yang symbol.

The cache block zones are interacted as follows: homebox-homebox,homebox-tower, homebox-yin, tower-yin, tower-yang. The inner loop takesadvantage of the packing of the cache blocks to merge these interactionsinto two composite interactions: homebox-(homebox,tower,yin) andtower-(yin,yang). It also sets up the inner loop such that no redundantpairs from the homebox-homebox interaction are considered. The onlymatchmaking criteria required is a distance check.

The zones are oriented such that the voxel strips used to encode theimport region run parallel to long direction of the two plate zones. Asa result, very few strips are needed to encode the import region. Theouter loop is arranged to traverse the homebox voxels such that theplate needed for each outer loop voxel significantly overlaps the platesneeded for the previous and next outer loop voxels, which improvestemporal locality.

The serialized computation described above can be combined with aparallel architecture, for example, using cache blocking techniques withthe processor at each node, while performing computations in parallel atdifferent nodes and communicating the results between the nodes afterthe computation is completed.

9 Cluster Architecture

Another approach to computation of particle interactions, which caninclude one or more of the algorithmic techniques described above, makeuse of a cluster architecture. In this architecture, a number ofcomputers are coupled by a communication network, for example by one ormore Gigabit Ethernet segments. Each computer can be a general purposecomputer, for example, executing an operating system (e.g., Linux).

In one example, each computer in the cluster is assigned a particularset of homeboxes of the simulation region, for example, assigningcomputers to homeboxes in a one-to-one manner. Each computer includesstorage for data for particles associated with its homebox. At the startof each iteration, the computers exchange particle data, for example,sufficient to implement a midpoint method for computing range-limitedinteractions. Other computations are also performed using this exchangeddata, for example, using the communication sharing approach described inSection 4. Some computations are performed in a distributed manner overmultiple computers. For example, FFT computations that are part of theGSE method can be implemented in a distributed manner.

10 Parallel Implementation

Another approach to computation of particle interactions, which caninclude one or more of the algorithmic techniques described above, makesuse of special-purpose hardware that is particularly suitable forvarious computations necessary for computing particle interactions. Thespecial purpose hardware is arranged in a parallel architecture thatincludes multiple processing nodes, with each processing node beingresponsible for the computations associated with one homebox. Thearchitecture permits internode communication. One arrangement of nodesis in a three-dimensional array in which each node communicates directlywith its six neighbors along three orthogonal axes, and communication ofparticle data can involve passing the data through multiple nodes toreach its destination.

Referring to FIG. 37, a representative three-dimensional array ofcomputation nodes 3720 is arranged such that each node performs thecomputation for a corresponding homebox 3710. The nodes are linked bycommunication paths 3730 along the orthogonal axes, and optionally bydiagonal communication paths 3735. Each processing node 3720 includes aprocessor 3720, which performs numerical calculations associated withthe particle interactions, and a communication interface 3750, which isresponsible for sending and receiving particle data over communicationlinks 3730, 3735 linking the node to its neighboring nodes.

Before discussing the details of the hardware implementation, thecomputation performed by the hardware can be understood with referenceto FIG. 38. The computation involves a loop that is performed at eachtime step of the simulation, with each time step representing asimulated time such as 1 or 2 femtoseconds (abbreviated fs, equal to10⁻¹⁵ seconds). Generally, each iteration involves three basic step.Velocity integration 3810 uses the calculated forces on each particle toupdate the velocity of that particle. Position integration 3820 uses theupdated velocities to calculate the updated positions of the particles.Force calculation 3830 uses the new positions of the particles tocalculate forces on the particles, thereby allowing the iteration torepeat.

Force calculation involves calculations at different spatial scales.These calculations are performed independently. There calculationsinclude bonded force calculation 3832, midrange force calculation 3834,and distant force calculation 3836. As part of each of thesecalculations, force accumulation 3838 determines the total force on eachparticle, with the force on any particular particle potentiallyincluding components that are determined by computations at each of thespatial scales. It is not necessary to perform the force calculations ateach of the spatial scales at each time step. The distant forcecomputations may be performed less frequently than the midrangecomputations, which are performed less frequently than the bonded forcecalculations, for example, using a MTS (Multiple Time Scale) and/orRESPA (REference System Propagator Algorithm) approach.

Another aspect of the parallel implementation involves dedicatingdifferent hardware elements to various types or spatial scales ofcomputations. For example, hardware elements are dedicated tocomputations involving pairwise particle interactions that are part ofthe midrange force calculation step 3834 (or particle-grid computationsthat form part of the distant force calculations 3836), and tocomputations of near interactions such that are part of the bonded forcecalculations 3832. In addition the force accumulation step 3838 makesuse of a memory system that incorporates numeric value accumulationfunctionality. In the multiple node architecture introduced above, eachnode includes such dedicated hardware elements for the various types ofcomputations.

Referring to FIG. 39, each computation node 3720 includes a number ofcomputation and communication elements. A flexible subsystem 3912includes one or more programmable processors. A midrange system 3914includes special-purpose circuitry that is adapted to perform pairwisecomputations, for example, between pairs of particles. An FFT (FastFourier Transform) subsystem 3924 is used in some embodiments to performcomputations related to distant force computations. Yet otherembodiments may include additional or fewer specialized or generalpurpose computations elements.

Communication within one processing node makes use of a ringcommunication architecture that includes a number of ring stations 3920coupled by communication links 3922. In this version of the system,there are eight ring stations per node, and the links are high speedserial communication links. Each ring station has additional links. Forexample links 3924 couple the ring stations to the computation elementsof the node. Other links 3730 couple ring stations for this node to ringstations 3930 of the six neighboring processing nodes 3720 of thetorroidal mesh of processors. Other links couple the ring stations to amemory controller 3950, which provides access to a memory 3954, and to ahost interface 3940, which is used for example for administrativefunctions such as loading programs to be executed on the computationelements.

The communication architecture, which is made up of the interconnectedcommunication rings on each node, supports a global addressing approachby which each computation or communication element coupled to a ringstation can address a data packet to any other element in the system,which may be on the same node, or may be on another node. Also, as isdescribed further below, the communication architecture supports formsof limited multicast that allows data packets sent from a source to bedistributed to multiple destinations without necessarily broadcastingthe packet to all nodes in the system or sending multiple copies of thedata packet from the source node.

In general, implementation of the iteration illustrated in the flowchartof FIG. 38 using the node architecture illustrated in FIG. 39 involvespartitioning tasks among nodes, and partitioning the processing stepsamong the computation elements of each node. Furthermore, each of thesubsystems takes advantage of internal parallelism. Overall, theimplementation provides parallel processing and pipelining ofcomputations at multiple levels of the architecture.

Referring to FIG. 40, computation elements for two nodes areillustrated, with indications of some of the data flows between theelements in a computation iteration. The communication links within anode and between nodes are not illustrated. The data flow that isillustrated relates to execution of the velocity and positionintegration steps (3810, 3820 in FIG. 38) as well as the midrange forcecalculation 3834. Other data flows, for example, related to distantforces and bonded forces are not illustrated in this figure.

Generally, at one point of the iteration time step, the flexiblesubsystem 3912 on each node retrieves particle force data for particlesit is responsible for from its own memory 3954. Using the force data, itperforms the integration steps required to determine the new particlepositions. This particle position data does not have to be returned tothe memories 3954. Rather, each flexible subsystem 3912 multicasts itsposition data, and this position data is passed to those midrangesystems 3914 that will use the data to compute pairwise particleinteractions, for example, using the NT method described above. Inparticular, the multicast approach ensures that the midrange subsystemof each node receives the particle data for the entire import regionneeded for the decomposition approach being used. Furthermore, data is“pushed” to the consumer of the data without necessarily requiringrequests or acknowledgements for the data, thereby reducingcommunication load and delay associated with such communication.

In a pipelined manner, the midrange subsystems 3914 begins computingpairwise interactions as they receive the required particle data. Forexample, the midrange subsystem 3914 at a node may receive the particleposition data for particles in that node's homebox relatively quicklyover on-node communication paths, and then start receiving position datafrom neighboring nodes.

As the midrange system 3914 computes force contributions for particles,it distributes these contribution terms to the memories 3954 at thenodes responsible for those particles. The memories include accumulationfunctionality that allows the memory on sum the force contributions oneach particle as the force contributions for that particle are receivedfrom various nodes. Using the accumulation function of the memories,different processors can compute components of a sum (e.g., of forces)and send them to a memory where the components are accumulated. Becausethe communication of the components does not require acknowledgementfrom the memories to the processors that produced the components,efficient accumulation is achieved. The memories implement theaccumulation so that it is not order-dependent so that the identicalresult is obtained regardless of the order in which components to besummed together arrive at the memory.

Not illustrated in FIG. 40 are similar pipelined parallel procedures forthe distance force computations, which involve grid-based computationsthat make use of Fast Fourier Transform (FFT) techniques that usemultiple interchanges of data between nodes, and in some embodimentsmakes use of the specialized FFT subsystem 3916 to implement the FFTcomputations.

Referring again to FIG. 39, the communication system supportspacket-based message interchange between elements on the same or ondifferent nodes. Packets can represent operations including read, write,accumulate, and read-modify-write with the packet header specifying theoperation and the packet body carrying operands (e.g., values to thewritten). A global address space is used for registers, memorylocations, etc. in any node, with a 48-bit address space having 16 bitsreserved for node number and the remaining 32 bits for 4G addresslocations (e.g., bytes) at each node. Different address ranges at a nodemap to different ring stations on the node based on routing informationstored in the ring stations, and at each ring station, different addressranges are mapped to different subsystems coupled to the ring station.Packets are variable length having between one (i.e., a header only) andnine units of length (a header plus eight units of data), with each unitbeing, for example, 256 bits.

Each ring station is programmable with respect to routing, and isconfigured for a particular routing pattern before the simulationiterations are started. Each ring station includes a configuration tablethat is loaded before processing begins. In some examples, this tablespecifies a set of production rules that are applied to each incomingdata packet. In general, the table in each ring stations specifies howpackets that arrive at that ring station should be handled. Each packetthat comes in is passed on one (or more) of the other ports (which cansend the data off to another node, or on its way to a module on the samenode) according to the table based on the source or destination address,and optionally based on a packet type. The configuration table can alsospecify an address translation function to be performed on the databefore it is forwarded. Every ring station in this embodiment has twoports for communication on the ring, and two client ports, for example,for connecting to ring stations at other nodes or for connecting tocomputation elements at the node.

Typically, messages that are sent from client to client are notacknowledged. However, for some communication an acknowledgement may begenerated by the destination element and used for synchronizationpurposes. This embodiment does not perform error correction if there arecommunication errors for example by retransmission. Rather, thecomputation is restarted from a previously checkpointed timestep.Checkpoints are generated periodically, for example, with a periodranging from approximately one second to one minute of real time.

One use of the multicast mechanism is for distribution of various typesof data to nodes according to the type of data. For example, dependingon the import region for particle data, the processors that need copiesof particle data from one processor is determined by the geometry of theimport region. The ring stations of the communication system areconfigured such that the source processor for the particle data does notneed to be aware to the communication interconnections across which theparticle data must pass, or where copies of the data are to be made.Rather the source processor tags the data with a group id correspondingto the type of data, and the communication system distributes it to allprocessors that need to import than data according to the configurationof the communication system.

A multicast mechanism is provided for distributing position data, forexample, in the implementation of the NT method discussed above.Multicast is implemented using a limited range of destination addresses,each corresponding to a different class of multicast communication,referred to as the multicast group id. A flag elsewhere in the header ofa packet indicates that the message is a multicast rather than a unicastpacket. When a multicast packet is received at a ring station, themulticast group id and the source node address, which is also in theheader, are examined and based on those values and on the address of thereceiving node, the packet may be replicated and sent on one or moreoutput ports of the ring station according to the ring stations routingtables. Therefore, a different pattern of multicast can be configuredfor each different multicast group id. Note that acknowledgements arenot required. The configuration of the multicast pattern is distributedamong the ring stations, each of which has a limited view of the overallrouting pattern.

Inside each client module (e.g., a computational subsystem), interfacehardware monitors “writes” from the ring station and based onpre-specified writes generates signals (e.g., flags, event codes) thatmay initiate appropriate computations within the client, for example, asa result of polling the generated signals.

In addition to the toroidal-mesh interconnect network, the system canoptionally have additional other interconnect topologies for differentpatterns of communication. For instance, certain global quantities suchas the temperature or pressure of the system may need to be computed.Global communication patterns may also be required to synchronize thenodes. Options for the additional topologies include a low-latency butnot necessarily high-bandwidth tree structure. In the presentimplementation, routing protocols are layered on top of the meshinterconnect to provide a virtual topology of the required type for thedifferent communication patterns.

Each subsystem at a node in general includes its own internal memories,which are separate from the memory 3954. The memory controller 3950includes a cache (e.g., that uses fast static RAM), which for manysimulation scenarios is sufficiently large to hold the data for all theparticles being handled by the node. The memory 3954 includes a largerstorage space, which may be DRAM, which can be used for larger problemswhere each node is responsible for more particles. The interchange ofdata between the SRAM cache and the DRAM is coordinated in a bufferingarrangement in which data in one buffer is being transferred while datain another buffer in the SRAM is being used for a computation.

For many of simulated systems (e.g., those of dimensions roughly 128{acute over (Å)} cubed, or smaller), it is expected that the memory 3954managed by the memory controller 3950 will not necessarily be used. Inthese cases, the primary purpose of the memory controller is theaccumulation of force data. The cache of the memory controller isdesigned for high bandwidth accumulation of force data produced by themidrange or distant interactions in the midrange subsystems and the bondterms and corrections produced by the flexible subsystems. The memorysubsystem of any specific node is responsible for accumulating forcedata for the particles owned by that node.

For large scale systems, the memory 3954 includes 1 to 2 GB of DRAM ateach node. In modes that use this DRAM, the system may be partitionedsystem is sub-partitioned—each box partitioned into sub-boxes—and thealgorithms described above are implemented on a sub-box by sub-boxbasis, with the required sub-boxes being pulled out of DRAM to thememory controller cache as necessary. Note also that such a sub-box modecan in fact be used on smaller systems, before data exceeds the capacityof the memory controller cache, and there may be cases where this yieldsa performance advantage.

The subsystems described above and shown in FIG. 39 cooperate in theevaluation of a force field, the parameters of which are provisioned atthe beginning of any simulation run. At the beginning of a simulation,the parameters of the force field, together with the description of thesystem—the particles (e.g., atoms) involved, their connectivity (e.g.,covalent bonds), their positions in space and their currentvelocities—are all known and partitioned appropriately among nodes ofthe system. Specifically, the particle data (positions and velocities)reside in the flexible subsystem of an particle's homebox. In addition,each bonded term has an “owner” node, responsible for its computation.

The flexible subsystem 3912 is the “choreographer” of each simulationtimestep. While most of the remaining subsystems are responsible forcomputation of the force field, the flexible subsystem implements theintegration algorithm, which computes accelerations from forces, andincrements velocities and positions incrementally from timestep totimestep. The integrator that is implemented for any particular type ofsimulation may require more than just simple integration: it may, forexample, implement a constant temperature simulation—which generallyrequires the periodic calculation of temperature throughout thesimulation. This in turn requires calculation of kinetic energy fromvelocity data, and the synthesis of a global scalar, the temperature,from this data. Such a particular computation, and others like it(pressure control, for example), are implemented by programming theflexible subsystem for these specific tasks; in this particular example,the flexible subsystems of all nodes must cooperate in computing aglobal reduction to compute this global scalar.

Before continuing with a detailed discussion of the overall computationtimestep, or the architecture of the flexible subsystem, it is useful toconsider the function and implementation of the midrange subsystem 3914.

The midrange subsystem 3914 can be thought of as a genericimplementation of the NT Method that supports a much broader range ofalgorithms and decompositions. The subsystem is designed to efficientlycalculate all pairwise interactions among point objects, whetherparticles or grid points, locally within a specified cutoff radius. Oneof its chief tasks is the computation of midrange non-bondedinteractions through full pairwise summation. It is also used in thereal-space components of the distant force calculation, for chargespreading and force interpolation.

Referring to FIG. 41, operation of the midrange subsystem can beunderstood by considering a single particle-particle interaction module(PPIM) 4100. Generally, for computation of interactions betweenparticles, during each iteration, memory in PPIM is first loaded withparticle data for one set of particles at the beginning of theiteration. Then, particles in another set are streamed past the PPIMproducing force contributions on the particles being streamed past. Atthe end of the iteration, force contributions for the particles thatwere resident in the PPIM is streamed out.

In some examples of computations performed by a PPIM, the PPIM operatesin three phases. In the first phase (indicated by the circled numeral 1in the figure), data for a first set of points, such as particles in the“tower” region for the NT method, are received over an input 4110 andstored in a memory 4130 of the PPIM. In a second phase, data for asecond set of points, such as for particles in the “plate” region forthe NT method, are received over input 4110 and passed to a computationelement 4140. For each received particle, the computation elementcomputes the total force (or other accumulation of pairwise computedquantities) on the particle from all stored particles, which were loadedin the first phase. This accumulated force is passed on an output 4122.In this second phase, there is one force send on output 4122 for eachparticle received on input 4110, with the ordering preserved. During thesecond phase, forces on the particles stored in memory 4130 are alsoaccumulated. In the third phase, these forces are sent on output 4122,with one force sent for each particle receive on input 4110 in the firstphase.

The PPIM includes storage for a number, for example three, differentcomputations. In this way, different computations can be performed bythe PPIM without reloading data, for example, allowing the first phase(data loading) phase of a second computation to be performed during thethird phase (result unloading) phase of a first computation.

Referring to FIG. 42, the midrange subsystem implements a degree ofparallelism by replicating the PPIM 4100 illustrated in FIG. 41 so thateach PPIM performs only a subset of the required computations. Ingeneral, during the first phase, each PPIM receives only a portion ofthe first set of particle data, and in the second phase receives all thedata in the second set. The outputs of all the PPIMs are then merged toyield the same result as would have been obtained by a single PPIMacting serially.

The midrange subsystem includes an array PPIMs 4100, as well as memorieson the input 4230 and output 4235 of the array. The PPIMs are arrangedinto multiple rows, each with multiple PPIMs. In the presentimplementation, each row has eight PPIMs and there are four rows, butother arrangements can also be used.

Considering first one row of PPIMs, each PPIM passes data on its input4110 (see FIG. 41) to its output 4112 without modification so that inthe second phase of computation, each PPIM in a row receives data forthe same set of particles. During the first phase, the data for theparticles in the first set are also passed from PPIM to PPIM, but eachPPIM stores only a separate portion of the data. During the secondphase, the output 4122 of each PPIM includes one accumulated force foreach particle input on its input 4110. At each PPIM (after the first,left-most) the PPIM receives a partially accumulated force for eachparticle on its input 4220, and internally sums the input contributionfor that particle with the contribution computed by its computationelement 4140, so that the output 4122 again has one force for eachparticle on its input 4110. Therefore, each row of PPIMs functions as asingle PPIM, but performs more computations in parallel and thereforeachieves higher throughput.

During the third phase, each PPIM streams out the accumulated forces foreach point in the first set that was loaded to that PPIM, with“downstream” PPIMs in the row passing that force data withoutmodification, with the overall order produced by the row matching theorder the particles were loaded into the PPIM row during the firstphase.

Considering the multiple rows of PPIMs, the data for the first set ofparticles is partitioned as it is read in so that each row has aseparate portion of the first set. The data is passed through a set ofdistribution units 4210 to achieve this partitioning. The result is thateach PPIM holds in its memory 4130 a different portion of the first setof points, with the PPIMs together holding all the points.

During the second phase, the distribution units 4210 replicate theparticle data such that the same sequence of particles are sent to eachrow of PPIMs, and therefore also to each PPIM in that row. This resultsin each row of PPIMs producing a force for each particle in the secondset, and these forces must be accumulated across the rows. Thisaccumulation is performed by the accumulation units 4220.

During the third phase, the accumulation units 4220 concatenate theforce data for the particles in the first set, so that at the output ofthe last accumulation unit 4220, the forces for the particles in thefirst set exit in the same order as the particle data was passed to thearray in the first phase.

The midrange subsystem includes a front-end memory 4320, which receivedthe particle data multicast from the flexible subsystems. A controller(not shown) monitors the writes to this memory and coordinates thepassing of data to the array when the required data is available. Thesubsystem includes a back-end memory 4235, which is used to stage thecomputed forces (or other data). Note that this data is sent to thememory subsystem without waiting for completion of the processing sothat some forces are being sent to the memory while other forces havenot yet been computed.

Referring to FIG. 43, each PPIM implements a further level ofparallelism. Each PPIM contains several matchmaking units (MUs) 4350. Inthe present implementation, each PPIM includes eight MUs. The particledata distributed to the PPIM during the first phase, described above asbeing stored in memory 4130 (see FIG. 41), is actually distributed tomemories 4332 in the MUs. That is, the particles in the first set arepartitioned among the MUs. During the second phase, all the particlesreceived by the PPIM are steamed past each matchmaking unit. For eachreceived particle in the second set, a computation element 4342 of an MUdetermines which points stored in that MU's memory 4332 shouldinteractions be computed, based primarily (but not exclusively) on thedistance between the particles. The computation element 4342 passes theselected pairs to a concentrator 4360, which receives such selectedpairs from each of the MUs, and acts essentially as an arbitrator forpassing the selected pairs in a serial manner to a particle-particleinteraction pipeline (PPIP) 4344. The PPIP then computes either theforce on each particle due to the other or the potential energycontribution from the pair. The PPIP can be configured to compute arather general function of the particles, based on their identities andinter-particle distance, and calculated by table-lookup andinterpolation. For example, the PPIP is implemented or preconfigured tosupport a number of different functional forms, with data loaded intothe PPIP determining the specific function within that functional form.The PPIP accumulates the forces for each particle in the second set fordifferent of the stored particles, and passes the accumulated forces onoutput 4122 while storing the forces on the stored particles in a memory4334. In the third phase, these stored forces in memory 4334 arestreamed out on output 4122.

In some implementations of the PPIP, the functions that can be computedby the PPIP are represented as piecewise analytic functions. That is,within each of a set of ranges of argument value, a parametric analyticfunction is used to determine the function of the argument. A datastructure stores the parameters of the analytic functions for thecorresponding different ranges of argument values. A variable precisionapproach is used such that in different sizes of ranges of argumentvalue can correspond to different parameters of the analytic function,for example, giving higher precision or greater accuracy in some rangesof argument value than in others. The selection of coefficients isperformed in two stages. First, the argument range is divided into Ncoarse ranges, each of which can have a different size. Each of thecoarse ranges is divided into the same number n of subranges, such thatwithin each coarse range the subranges have the same size. For aparticular input argument, first the coarse range is determined, andthen the subrange within that coarse range is determined in order tolook up the appropriate parameters of the analytic function. Therefore,there are N times n sets of coefficients. Given a limited number of setsof coefficients that can be used, the accuracy of the function to anideal function can be chosen according to the argument range. An exampleof such a ideal function is erfc(1/R) and an example of the analyticfunctions are quadratic functions.

The PPIM array of the midrange subsystem in also used for chargespreading and force interpolation computations of the GSE methodsdescribed earlier in this document. For both charge spreading and forceinterpolation (the first and last thirds of the distant computation)particles are distributed to the PPIMs in a first phase, and the gridlocations are streamed past the PPIMs in a second phase. Note that gridlocations for the homeboxes of neighboring nodes streamed, as well asgrid locations of the current node's homebox. Note that for thiscomputation using an NT method, a “full plate” is needed because thereis an ordering for the particle interaction so two equal and oppositeforces are not computed at one time. During the charge spreading, nodata needs to be accumulated by the PPIMs and streamed out in the thirdphase, while in force interpolation, the PPIMs do not need to provideoutputs during the second phase, with the interpolated forces beingaccumulated during the second phase and streamed out of the PPIMs in thethird phase.

The flexible subsystem 3912 also implements parallelism within thesubsystem. Referring to FIG. 44, a groups of computational elements 4410function relatively independently in parallel. Not only is each particleassigned to a particular node within the system (i.e., the nodeassociated with the homebox holding the particle) each particle isassigned to a particular group 4410 within the flexible subsystem ofthat node. In the present implementation, there are four groups ofcomputation elements 4410.

Each group 4410 includes a processor 4414. Each processor has data andinstruction caches (32 KB) 4412 that are backed by the memory systemaccessed over the communication ring. In many applications the caches4412 are sufficiently large to be able to hold all the data needed foran iteration. Each processor has a remote access unit (RAU) 4418, whichimplements functions including direct memory access (DMA) functions fortransferring data from the memory over the communication ring to amemory 4416, which is accessible to the processor 4414. The particledata that is the responsibility of processor is held in memory 4416, andthe RAU 4418 retrieves force data for those particles from the memorysubsystem using a DMA approach without loading the processor 4414. TheRAU is coupled to a ring station through a communication interface,which aggregates outputs and distributes inputs to the flexiblesubsystem to the appropriate elements (e.g., based on the addresses ofreceived packets).

Each processor has a number of slave special-purpose co-processors 4420,referred to below as “geometry cores” (GCs). In the presentimplementation, there are two GCs per processor. Each GC has a four-waySIMD architecture, dual issue VLIW pipelined ALU, and includes“geometric” primitives including computation of inner (“dot”) products.Each GC uses a vector/scalar register architecture in which registerscan be addressed in an interleaved manner as either n vector registersor as 4n scalar registers, and where scalar results can be placeddirectly in a portion of a register where needed for subsequentcomputations. There are inbound queues for the GCs that are fed by itsprocessor, and outputs from the GC can be fed back to the memory systemwithout requiring handling by the processor.

The GCs are used to compute bonded force terms. Each bond is assigned toa particular GC, and the processors exchange particle position data sothat each processor has the data it needed to have its GC compute bondedforce contributions. Note that a processor that computes a bonded forcecontribution may not be responsible for any of the particles in thatbonded pair. Periodically (e.g., every 2000 time steps) a reassignmentof bonds to processors and GCs may be performed as an optimization thatmay reduce communication requirements.

In a computation iteration, particle position data is distributed amongthe memories 4416 of the processors of the flexible subsystems of thenodes of the overall system. From there, the particle locations are sentto the midrange subsystems for computation of the pairwise interactions,and to other processors for computation of bonded terms.

The flexible subsystem 3912 also includes a correction processor 4430which is used to add correction terms to values computed by the GCs. Thecorrection processor implements an example of a general computationalapproach. In its general form, if one computation is to be performed ona large part A of a data set and a second computation is to be performedon a small part b of the data set, where A and b form a partition of thecomplete data set, and the results of the computations are to beaccumulated (or more generally reduced) over the data set, then it maybe efficient to perform and accumulate the first computation over theentire data set, and then for the small part b recompute and subtractthe result of the first computation and compute and add the result ofthe first computation. For example, by not having to make exceptions forelements of the data set in part b, the first computation may be madeefficient by regular (i.e., condition or exception free) iteration. Inthe context of particle interactions the entire data set (i.e. A plus b)can correspond to a set of particle pairs that are within an interactionradius, the first computation can be a unbonded interaction, forexample, as computed in the PPIM array, and the second computation canbe a bonded interaction as computed in the GCs. So rather than having toexclude bonded pairs of particles that are within the interaction radiusfrom force computations in the PPIM array, when a bonded interactionbetween particles is computed a GC, the correction processor 4430computes the negative to the unbonded term that is computed in the PPIMfor that pair of particles, and adds it to the bonded term before it issent to the memory where it is accumulated. In some implementations, thecorrection processor includes the same computational structure as a PPIP4344, further ensuring that exactly the same unbonded term is computedfor subtraction from the total.

There are generally three major computations that are hosted in theflexible subsystem: bonded force terms, constrain terms and integration(performed without the constraints), and finally updated positionsaccounting for the constraints. During the computation of distantforces, the processors and the GCs also perform performing the FFTcalculations in versions of the system that do not include a dedicatedFFT subsystem.

In versions of the system that use the flexible subsystem to perform theFFT computations, the distant forces are computed by first distributingcharges to the grid locations using the midrange subsystem. Thesedistributed charges are returned to the flexible subsystem, whichimplements a distributed FFT computation for the grid spanning theentire simulation volume. The FFT is distributed among the processors(and their associated GCs) and requires multiple phases of exchange ofintermediate results between the nodes. At the end of the FFT iteration,the transformed data, which represents potentials at grid locations, ispassed back to the midrange subsystem, where it is used to computeinterpolated forces on the particles. These interpolated forces are thensent to the memory subsystem whether the distant force terms arecombined with the other force terms computed for each particle.

In the present system, each node is implemented using an applicationspecific integrated circuit (ASIC) that includes all the computation andcommunication subsystems shown in FIG. 39, with the exception of thememory 3954 and remote ring stations 3930. Multiple ASICs are arrangedone each of multiple circuit boards that are interconnected through abackplane. The internode communication links 3730 are therefore eitherlocal to a circuit board or are routed through the backplane.

11 Applications

The algorithmic techniques, or the hardware or software techniques,described above may be applicable to a variety of applications, some ofwhich are related to biomolecular simulation while others are not. Theapplications may include, for example, the following:

Biomolecular simulation applications related to use of moleculardynamics, Monte Carlo simulation, or other sampling technique, caninclude without limitation: ab initio protein structure prediction,refinement of homology models of protein structure, modeling of proteinfolding kinetics, computation of protein-ligand binding free energies,determining whether a pair of proteins will interact; determining theirbinding site and their binding free energy; toxicology studies;computation of hydration free energies; protein design; modelingallosteric changes and protein function; simulation of membraneproteins; simulation of nucleic acids; simulation of carbohydrates;simulation of combinations of the above and interactions between them;predicting membrane penetration, including blood-brain barrierpenetration; molecular force field development; and simulations ofcrystal structures

Applications may also involve other fields, such as material scienceapplications, weather simulations, hydrodynamic simulations, equation ofstate simulations, astrophysics simulations, etc.

12 Alternatives

A variety of arrangements of distributed processing nodes canalternatively be used with the computation decompositions describedabove.

It is also not necessary that there be a one-to-one correspondencebetween processing nodes and homeboxes. For example, each node may beresponsible for a set of homeboxes, such as a n×n×n cube of homeboxes.One use of such a finer-grain definition of homeboxes is inmultiple-zone scheduling approaches in which the schedule ofzone-to-zone interactions can be used to avoid a need to test forduplicate computations or can be used to reduce the amount ofcommunication through improved rounding. Another use of such finer-grainhomeboxes is in serializing the computation involved with each pair ofinteracting zones to make better use of local memory, such as localcache memory, at each of the processing nodes. Aspects of memoryutilization are discussed further below in the context of serialimplementations.

In one alternative version of a parallel implementation, each nodemaintains data for more particles than those that are located within itshomebox. A “clone buffer” approach is used in conjunction with amidpoint assignment rule approach. In particular, each node storesparticle data from a region that extends a distance R/2 beyond theboundaries of its homebox. After each computation phase, the forces forall the particles in the extended region are sent to that node allowingit to redundantly compute the new locations of the particles. During thecomputation phase, further importing of particle data is not necessary,for example, using the midpoint assignment rule or the pseudo-randomassignment rule, because the extended region includes all the particlesneeded.

The approaches described above can be implemented in software, inhardware, or using a combination of hardware and software. For example,alternative versions of the system can be implemented in software, infirmware, in digital electronic circuitry, in computer hardware, inother modes of implementation, or in combinations of such modes. Thesystem can include a computer program product tangibly embodied in amachine-readable storage device, such as on a magnetic or an opticaldisc or in a semiconductor memory, for execution by a programmableprocessor or by a system that includes multiple programmable processors.The programmable processors can be general-purpose processors, or can bespecial-purpose processors, for example, designed for computations ofthe types described above. Parallel implementation can, for example, usespecial-purpose interconnection networks, or can use more generalpurpose data networking approaches. Method steps can be performed by aprogrammable processor executing a program of instructions to performfunctions by operating on input data and generating output. The systemcan be implemented in one or more computer programs that are executableon a programmable system including at least one programmable processorcoupled to receive data and instructions from, and to transmit data andinstructions to, a data storage system, at least one input device, andat least one output device. Each computer program can be implemented ina high-level procedural or object-oriented programming language, or inassembly or machine language if desired. Such language can be either acompiled or interpreted language. Suitable processors include, by way ofexample, either (or both) general and special purpose microprocessors.Generally, a processor will receive instructions and data from read-onlymemory and/or random access memory. Generally, a computer will includeone or more mass storage devices for storing data files. Such devicesinclude magnetic disks such as internal hard disks and removable disks,magneto-optical disks, or optical disks. Storage devices suitable fortangibly embodying computer program instructions and data include allforms of non-volatile memory, including by way of example semiconductormemory devices such as EPROM, EEPROM, and flash memory devices; magneticdisks such as internal hard disks and removable disks; magneto-opticaldisks; and CD-ROM disks. Any of the foregoing can be incorporated in orsupplemented application-specific hardware, such as application-specificintegrated circuits (ASICs).

Other embodiments are also within the scope of the appended claims.Reference numerals in the claims, which are provided in parentheses toincrease intelligibility, should not be construed as limiting the claim.

1. A method for performing computations associated with bodies locatedin a computation region, the method comprising: for each subset ofmultiple subsets of the computations, performing the computations inthat subset of computations, including accepting data of bodies locatedin each of a plurality of import regions associated with the subset ofthe computations, the import regions being parts of the computationregion; for each combination of a predetermined plurality ofcombinations of multiple of the import regions, performing computationsassociated with sets of bodies, wherein for each of the sets of bodies,at least one body of the set is located in each import region of thecombination.
 2. The method of claim 1, wherein at least some of thecombinations of multiple of the import regions each consists of a pairof the import regions.
 3. The method of claim 1, wherein each of thesubsets of computations is performed on a different node of a multiplenode computing system.
 4. The method of claim 3, wherein accepting thedata for bodies from the import regions includes accepting data from acommunication medium coupling at least two of the nodes of the computingsystem.
 5. The method of claim 1, wherein each of the subsets ofcomputations is performed in sequence on a computing system.
 6. Themethod of claim 5, wherein the subsets of computations are selectedaccording to memory limitations of the computing system.
 7. The methodof claim 1, further comprising associating each of the subsets ofcomputations with an iteration in an iterative sequence of computationphases.
 8. The method of claim 7, wherein accepting the data for bodiesfrom the import regions includes loading the data into a local memoryassociated with a processor on which the computations for the iterationare performed.
 9. The method of claim 1, wherein each of the subsets ofcomputations is associated with a separate sub-region of the computationregion.
 10. The method of claim 9, wherein for each of the subsets ofcomputations, the import regions are separate from the sub-region of thecomputation region with which the subset is associated.
 11. The methodof claim 9, wherein the sub-regions associated with the multiple subsetsof computations form a partition of the computation region.
 12. Themethod of claim 9, wherein the sub-regions form a regular partition ofthe computation region.
 13. The method of claim 12, wherein thesub-regions form a parallelepiped partition of the computation region.14. The method of claim 9, wherein performing the computations in eachsubset of computations further includes: performing computationsassociated with sets of bodies, for each of the sets each body of theset being located in the sub-region associated with the subset ofcomputations; and performing computations associated with sets ofbodies, for each of the sets one body being located in the sub-regionassociated with the subset of computations and the other bodies of theset being located in the import regions associated with the subset ofcomputations.
 15. The method of claim 14, wherein the performing ofcomputations associated with sets of bodies in which each body of theset of bodies is located in the sub-region associated with the subset ofcomputations is performed while accepting the data for bodies from theimport regions.
 16. The method of claim 9, wherein performing thecomputations in each subset of computations further includes: performingcomputations associated with sets of bodies, each body of the set beinglocated in one of the import regions associated with the subset ofcomputations, and no body of the set being located in the sub-regionassociated with the subset of computations.
 17. The method of claim 1,wherein performing the computations in each subset of computationsincludes applying a schedule of computations specified according to theimport regions.
 18. The method of claim 1, wherein each of thecomputations in the set of computations includes a computation of datacharacterizing an interaction of the set of bodies associated with thatcomputation.
 19. The method of claim 1, wherein each of the computationsin the set of computations includes recording in a data structure anidentification of the set of bodies associated with that computation.20. The method of claim 19, further comprising performing computationsof data characterizing an interaction of each of the sets of bodiesrecorded in the data structure.
 21. The method of claim 1, wherein inperforming the computations in all the subsets of computations, for anyset of bodies, a computation for that set of bodies is performed in atmost one of the subsets.
 22. The method of claim 1, wherein for at leastsome of the selected plurality of combinations of multiple importregions, a computation is performed for every combination of one body ineach of the multiple import regions.
 23. The method of claim 22, whereinin performing the computations in all the subsets of computations, forany set of bodies, a computation for that set of bodies is performed inat most one of the subsets.
 24. The method of claim 1, wherein the stepsare repeated for each of a series of time steps, the computationsperformed at each time step being used to update locations of the bodiesin the computation region for use in a subsequent time step.
 25. Amethod for performing a set of computations associated with bodieslocated in a computation region, each of the computations in the set ofcomputations being associated with a pair of the bodies, the methodcomprising: accepting data for bodies located in a neighborhood, theneighborhood comprising a plurality of zones; and performingcomputations associated with pair of bodies, the bodies of the pairbeing located in different of the zones; wherein a spatial extent of atleast one of the zones is determined to eliminate at least some pointsin one of the zones that are further away than a minimum distance fromall points in another of the zones.
 26. The method of claim 25, whereinthe spatial extent is determined according to a threshold distancebetween pairs of bodies for which computations are to be performed. 27.The method of claim 25, wherein a spatial boundary of one of theplurality of zones is determined according to a minimum distance topoints within another of the plurality of zones.
 28. The method of claim25, wherein at least one of the plurality of zones includes a non-planarboundary.
 29. The method of claim 25, wherein the computation region ispartitioned into regular regions, and at least one of the plurality ofzones includes some but not all of one of the regular regions.
 30. Themethod of claim 25, wherein the computation region is partitioned intoregular regions and each region is associated with a subset of thecomputations.
 31. The method of claim 30, wherein the neighborhoodcorresponds to one of the regular regions.
 32. The method of claim 31,wherein each regular region comprises one of the zones of theneighborhood corresponding to the regular region.
 33. The method ofclaim 31, wherein each regular region is associated with a correspondingcomputation unit, and wherein accepting the data for bodies in theneighborhood corresponding to the regular region comprises acceptingdata for one or more of the zones of the neighborhood from anothercomputation unit.
 34. A method for performing a set of computationsassociated with bodies located in a computation region, the computationregion being partitioned into regular regions, each of the computationsin the set of computations being associated with a set of the bodies,the method comprising: for at least some of the regular regions,accepting data for bodies located in a neighborhood of the regularregion, the neighborhood comprising a plurality of zones; wherein theplurality of zones includes at least one zone that is not contiguouswith parts being distributed in at least two dimensions of thecomputation region.
 35. The method of claim 34, wherein the plurality ofzones includes at least one zone that is not contiguous with parts beingdistributed in at least three dimensions of the computation region. 36.The method of claim 34, wherein each computation in the set ofcomputations is associated with a pair of bodies.
 37. The method ofclaim 36, wherein the neighborhood consists of two zones, and for eachof the pairs of bodies, one body of the pair is located in a differentone of the two zones.
 38. The method of claim 34, wherein the pluralityof zones includes at least one contiguous zone.
 39. The method of claim38, wherein the contiguous zone is contiguous with the regular region.40. The method of claim 34, wherein at least two of the zones areelongated in one dimension of the computation region.
 41. The method ofclaim 40, wherein at least one of the zones extends over at least oneother dimension of the computation region.
 42. A method for performingcomputations associated with bodies located in a computation regionusing a processor coupled to a local memory, the method comprising:iteratively performing multiple subsets of the computations, in eachiteration performing one subset of the multiple subsets of computations,including loading into the local memory data of bodies located in eachof a plurality of import regions associated with the subset ofcomputations, the import regions being parts of the computation region;performing on the processor computations associated with pairs ofbodies, wherein for each pair, one body of the pair is located in afirst of the import regions and the other body of the pair is located ina second of the import regions.
 43. The method of claim 42 wherein theperforming the computations associated with pairs of bodies is performedfor each of a predetermined plurality of combinations of a first and asecond of the import regions.
 44. The method of claim 42 whereinperforming the computations associate with the pairs of bodies includesperforming computations associated with substantially all pairs ofbodies in which one body of the pair is located in the first of theimport regions and the other body of the pair is located in the secondof the import regions.
 45. The method of claim 42 wherein the subsets ofcomputations are selected according to a limitation of the local memory.46. A method for performing computations associated with bodies locatedin a computation region, the method comprising: for each of a pluralityof sub-regions of the computation region, maintaining data for bodies inthat sub-region in a corresponding separate storage area, wherein thesub-regions overlap such that at least some bodies have data maintainedin multiple of the storage areas; and for each of the sub-regions,accessing the data for bodies in that sub-region from the correspondingstorage area and performing computations associated with sets of two ormore of the bodies in that sub-region.
 47. The method of claim 46wherein each of the separate storage areas is associated with a node ofa multiple node computing system.
 48. The method of claim 47 whereinperforming the computations associated with pairs of the bodies in thesub-region is performed on the node associated with the storage areacorresponding to the sub-region.
 49. The method of claim 47 wherein eachof the separate storage areas is associated with a different node. 50.The method of claim 46 wherein each of the sub-regions includes a homeregion and a region extending at least a predetermined distance from thehome region.
 51. The method of claim 50 wherein the home regions form apartition of the computation region.
 52. The method of claim 50 whereinthe predetermined distance is at least half an interaction radius forpairwise interactions between bodies.
 53. The method of claim 52 whereinperforming the computations associated with the pairs of bodies in thesub-region includes performing computations for pairs of bodies forwhich a midpoint of locations of the bodies is within the home regionfor the sub-region. 54-130. (canceled)
 131. A method of simulatingphysical interactions between bodies having locations in a computationregion, said method comprising: dividing said computation region into aplurality of import regions, each of which contains bodies; selecting,from said import regions, a first import region having a first body setand a second import region having a second body set; identifyingcombinations of bodies, each of said combinations including a body fromsaid first body set and a body from said second body set; from saididentified combinations, identifying a first combination set, whereinbodies associated with each combination in said first combination setsatisfy a condition; and identifying a second combination set, whereinbodies associated with each combination in said second combination setfail to satisfy said condition; for all of said identified combinations,other than said combinations in said first combination set, performingcomputations for simulating interactions between bodies in saidcombination, whereby computations to simulate interactions betweenbodies in combinations that satisfy said condition are avoided and onlycomputations to simulate interactions between bodies in combinationsthat fail to satisfy said condition are carried out.
 132. A method ofsimulating physical interactions between bodies having locations in acomputation region, said method comprising: dividing said computationregion into a plurality of import regions, each of which containsbodies; selecting, from said import regions, a first import regionhaving a first body set and a second import region having a second bodyset; identifying combinations of bodies, each of said combinationsincluding a body from said first body set and a body from said secondbody set; from said identified combinations, identifying a firstcombination set, wherein bodies associated with each combination in saidfirst combination set satisfy a condition; and identifying a secondcombination set, wherein bodies associated with each combination in saidsecond combination set fail to satisfy said condition; performingcomputations for simulating interactions between bodies in saididentified combinations, wherein performing said computations includesperforming computations for simulating interactions between bodies insaid combinations from said second combination set; and omittingperformance of computations for simulating interactions between bodiesin said combinations from said first combination set; wherebycomputations to simulate interactions between bodies in combinationsthat satisfy said condition are avoided and only computations tosimulate interactions between bodies in combinations that fail tosatisfy said condition are carried out.